Science, Faculty of
Mathematics, Department of
DSpace
UBCV
Hughes, Thomas J.
Doctor of Philosophy - PhD
University of British Columbia
We establish new fractal properties for superprocess densities. Consider the density of super-Brownian motion in one dimension at a fixed time. We construct a random measure supported on the boundary of the zero set of the density, which we call the boundary local time, and prove that it is locally non-degenerate, in the sense that it almost surely has positive mass on every neighbourhood of the boundary of the support. We then use the boundary local time to give an almost sure lower bound on the Hausdorff dimension of the boundary of the zero set, confirming a conjecture of Mueller et al. [69], whose work on this problem motivated ours. The Hausdorff dimension is characterized in terms of the lead eigenvalue of a certain killed Ornstein-Uhlenbeck operator which also appears in the construction of the boundary local time. The proof of local non-degeneracy includes a novel analysis of the density near the right endpoint of its support, including a new bound on the expected mass in a neighbourhood of the endpoint. We then consider the density of (α, β)-superprocess in dimension d with α â (0, 2), β â (0, 1], and βd < α. A classical result called instantaneous propagation asserts that, at a fixed time, the density has positive mass on any open set almost surely when conditioned on survival. We give general conditions under which the integral the density against a measure μ is positive almost surely when conditioned on survival. In particular, we show that this is determined by the relationship between the dimension of the measure, which we define in two ways, and a critical parameter determined by α, β and d. When the density does not charge μ almost surely, we obtain upper and lower bounds on the probability that it does so. In both cases our results correspond to new existence and non-existence results for solutions to the partial differential equation which is dual to the process. In the regime where the density is continuous and positive at a fixed point almost surely, we prove that the density function is strictly positive.
A boundary local time for super-Brownian motion and newpath properties for superprocess densitiesbyThomas J. HughesB.Sc., Queenâs University, 2014M.Sc., University of British Columbia, 2016a thesis submitted in partial fulfillmentof the requirements for the degree ofDoctor of Philosophyinthe faculty of graduate and postdoctoral studies(Mathematics)The University of British Columbia(Vancouver)August 2020c© Thomas J. Hughes, 2020The following individuals certify that they have read, and recommend to the Faculty ofGraduate and Postdoctoral Studies for acceptance, the thesis entitled:A boundary local time for super-Brownian motion and new path propertiesfor superprocess densitiessubmitted by Thomas J. Hughes in partial fulfillment of the requirements for the degreeof Doctor of Philosophy in Mathematics.Examining Committee:Edwin Perkins, MathematicsSupervisorMartin Barlow, MathematicsSupervisory Committee MemberJun-cheng Wei, MathematicsUniversity ExaminerNick Harvey, Computer ScienceUniversity ExaminerAdditional Supervisory Committee Members:Gordon Slade, MathematicsSupervisory Committee MemberiiAbstractWe establish new fractal properties for superprocess densities.Consider the density of super-Brownian motion in one dimension at a fixed time. Weconstruct a random measure supported on the boundary of the zero set of the density, whichwe call the boundary local time, and prove that it is locally non-degenerate, in the sense thatit almost surely has positive mass on every neighbourhood of the boundary of the support.We then use the boundary local time to give an almost sure lower bound on the Hausdorffdimension of the boundary of the zero set, confirming a conjecture of Mueller et al. [69],whose work on this problem motivated ours. The Hausdorff dimension is characterizedin terms of the lead eigenvalue of a certain killed Ornstein-Uhlenbeck operator which alsoappears in the construction of the boundary local time. The proof of local non-degeneracyincludes a novel analysis of the density near the right endpoint of its support, including anew bound on the expected mass in a neighbourhood of the endpoint.We then consider the density of (α, β)-superprocess in dimension d with α â (0, 2), β â(0, 1], and βd < α. A classical result called instantaneous propagation asserts that, at afixed time, the density has positive mass on any open set almost surely when conditioned onsurvival. We give general conditions under which the integral the density against a measureµ is positive almost surely when conditioned on survival. In particular, we show that thisis determined by the relationship between the dimension of the measure, which we definein two ways, and a critical parameter determined by α, β and d. When the density doesnot charge µ almost surely, we obtain upper and lower bounds on the probability that itdoes so. In both cases our results correspond to new existence and non-existence resultsfor solutions to the partial differential equation which is dual to the process. In the regimewhere the density is continuous and positive at a fixed point almost surely, we prove thatthe density function is strictly positive.iiiLay SummaryWe study two models for the density of a random spatial population. In the first model,individuals move locally in space and produce zero or two offspring in each generation. Weconstruct a new tool for quantifying the interface between the region where the populationdensity is zero and the region where it is positive and use it to compute the dimension (andhence size) of the interface.In the second model, individuals move in space via potentially long-range jumps and cancreate large numbers of offspring simultaneously. In this case, we characterize the regionson which the density is locally positive in terms of the dimension of the region. In a specialcase, we show that the density is positive everywhere in space. These probabilistic resultsallow us to extend recent results in deterministic partial differential equations.ivPrefaceThis thesis is based on three articles, two of which are published and one of which has beensubmitted for publication. Some minor changes have been made to the published articles.A version of Chapter 2 has been published as A boundary local time for one-dimensionalsuper-Brownian motion, of which I am the only author. It is published in The ElectronicJournal of Probability, Volume 24, paper no. 54.A version of Chapter 3 has been published as The boundary of the zero set of super-Brownian motion and its local time, by Ed Perkins (my thesis advisor) and myself. Thiswork is published in Annales de lâInstitut Henri Poincare´ B: Probabilite´s et Statistiques,Volume 55-4. Our relative contributions to research and writing are as follows: two keyauxiliary results are Proposition 3.3.5 and Proposition 3.3.6; the former is due to me andthe latter is due to Ed Perkins. The remaining sections are the product of collaborativeresearch. I prepared the first typed draft of Sections 3.4 and 3.5, and Ed Perkins preparedthe first typed draft of Section 3.1. Everything not mentioned above was prepared jointly.Chapter 4 is unpublished, original work of which I am the only author, and has beensubmitted for publication under the title The density of the (α, β)-superprocess and singularsolutions to a fractional non-linear PDE.Chapter 1 is an original introduction to the field, which I wrote, and includes somematerial from the other chapters.The work in Chapters 2 and 3 is a continuation of a research program initiated byMueller, Mytnik and Perkins (2017), and some results contained there were motivated by(and are resolutions to) conjectures of those authors. Methodologically, these chaptersinclude both novel techniques and extensions of the existing methods, some developedindependently by me and some in collaboration with my advisor. I independently developedthe research program of Chapter 4.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 A critical branching particle system . . . . . . . . . . . . . . . . . . . . . . 11.2 Continuous state branching and superprocesses . . . . . . . . . . . . . . . . 31.2.1 Warm up: a limit theorem for branching processes . . . . . . . . . . 31.2.2 Continuous state branching with general branching mechanisms . . . 51.2.3 Spatial branching particle systems, superprocesses, and their duals . 111.3 Two superprocesses and their properties . . . . . . . . . . . . . . . . . . . . 151.3.1 Super-Brownian motion . . . . . . . . . . . . . . . . . . . . . . . . . 151.3.2 The (α, β)-superprocess . . . . . . . . . . . . . . . . . . . . . . . . . 161.3.3 Super-Brownian motion with (1 + β)-stable branching . . . . . . . . 171.3.4 Supports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.3.5 Densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.3.6 The canonical measure and cluster decompositions . . . . . . . . . . 241.4 Our Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261.4.1 A boundary local time for super-Brownian motion . . . . . . . . . . 271.4.2 New path properties for the (α, β)-superprocess . . . . . . . . . . . . 331.5 Notes on the text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38vi2 The boundary local time of super-Brownian motion I: construction andproperties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.1 Introduction & statement of main results . . . . . . . . . . . . . . . . . . . 412.2 Killed Ornstein-Uhlenbeck processes . . . . . . . . . . . . . . . . . . . . . . 512.3 Some non-linear PDE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602.4 Existence and properties of Lt . . . . . . . . . . . . . . . . . . . . . . . . . . 652.5 Proof of Theorem 2.1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 832.5.1 PDE representations and preliminary bounds . . . . . . . . . . . . . 832.5.2 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 942.6 Proof of Lemma 2.5.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1033 The boundary local time of super-Brownian motion II: 0-1 law . . . . . 1173.1 Introduction and statement of results . . . . . . . . . . . . . . . . . . . . . . 1183.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1253.2.1 Killed Ornstein-Uhlenbeck processes . . . . . . . . . . . . . . . . . . 1253.2.2 Cluster and historical decompositions of super-Brownian motion . . 1253.2.3 Hitting probabilities of super-Brownian motion . . . . . . . . . . . . 1273.3 Some semi-linear partial differential equations . . . . . . . . . . . . . . . . . 1283.4 Proof of Theorem 3.1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1413.5 Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1524 New properties for the density of the (α, β)-superprocess . . . . . . . . . 1554.1 Introduction and statement of results . . . . . . . . . . . . . . . . . . . . . . 1564.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1714.2.1 Transition densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1714.2.2 The density of the (α, β)-superprocess . . . . . . . . . . . . . . . . . 1724.2.3 The fractional PDE and µ(Xt) . . . . . . . . . . . . . . . . . . . . . 1764.2.4 A Feynman-Kac formula . . . . . . . . . . . . . . . . . . . . . . . . . 1814.2.5 Cluster decompositions . . . . . . . . . . . . . . . . . . . . . . . . . 1814.3 The density at a fixed point . . . . . . . . . . . . . . . . . . . . . . . . . . . 1824.4 Strict positivity of the density . . . . . . . . . . . . . . . . . . . . . . . . . . 1884.5 Almost sure charging of (F1)-s measures when β â¤ Î²â(Î±, s) . . . . . . . . . 1934.6 Decay of Nx(µ(Xt) > 0) for (F2)-s measures when β > Î²â(Î±, s) . . . . . . . 1984.7 The initial trace problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2085 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2135.1 Super-Brownian motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2135.2 The (α, β)-superprocess . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214viiBibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217viiiAcknowledgmentsFirst and foremost, I would like express my deep gratitude to my advisor, Ed Perkins, forhis guidance, patience and encouragement over the past six years. Ed introduced me to thefield, and his conscientious supervision kept me on track while allowing me to flourish andexplore. His mentorship has been a great influence on me, mathematically and personally,and his generosity and wit are a joy and inspiration to those around him. Thank you, Ed.I thank Martin Barlow and Gordon Slade for serving on my supervisory committee,Jun-cheng Wei and Nick Harvey for serving as university examiners, and Vitali Wachtel forserving as the external examiner. Thank you for lending your time and expertise.I am grateful to the members of the UBC Mathematics Department for providing suchan excellent working environment â the faculty, the staff, my fellow students, and myforever office mate Sara´ı. I am lucky to have been a member of UBCâs probability group,and I thank the past and present members of this group for the stimulating conversations,seminars and reading groups from which I learned so much. Thanks are also due to theorganizers of the various schools and conferences which I have attended and from which Ihave benefited greatly. I am especially grateful to Alison Etheridge for generously invitingme to visit Oxford, and to Leonid Mytnik and Yaozhong Hu for inviting me to conferencesat BIRS and the UofA, respectively.Thanks to many friends old and new, near and far, mathematicians and otherwise, whohave been there over the years. To my family: thank you for your support, which has beena constant comfort throughout my studies. And to Claire: thank you for everything â muchof my share of happiness and success I owe to you.Finally, I gratefully acknowledge the financial support I have received from NSERC,UBC, and the Li Tze Fong Memorial Fellowship.ixTo Claire and my family.xChapter 1IntroductionA Dawson-Watanabe superprocess, or simply a superprocess, is a measure-valued Markovprocess which models a spatial population evolving in time. The evolution has two essen-tial sources of randomness: random spatial motion and random reproduction (branching).Critically, these degrees of randomness are independent, as are the evolutions of differentindividuals in the population.In this Introduction, we first consider the motivating example of a continuous-timebranching particle system in d-dimensional Euclidean space, Rd. This is followed by ageneral discussion of non-spatial branching processes and their scaling limits, continuousstate branching processes, after which we reintroduce spatial motion in order to definegeneral superprocesses. We then specialize our discussion to the processes considered inthis thesis and review their known properties before discussing our own contributions.1.1 A critical branching particle systemWe wish to describe a branching particle system whose heuristic description is as follows:a collection of particles undergo independent random motion in space; periodically, eachindividual has a reproduction event in which the individual (the parent) dies and is replacedwith a random number of new individuals (the offspring) at the parentâs location. Thedistribution of the offspring is such that the mean number of offspring from each birthevent equals 1. This condition is known as criticality and ensures that the expected numberof individuals in the population is constant and equals the initial size of the population.For the present case, we assume that branching is binary: the parent will have either 0 or2 offspring, each with probability 12 .We fix a Markov process with state space Rd and denote its paths by (Î¾t)tâ¥0. ThisMarkov process is the spatial motion of the branching particle system. We fix N â N anddefine the set of branch times { 1N , 2N , . . .}. Now, consider a population which begins, at1time 0, with N individuals located at the origin. Each individual evolves in space as anindependent copy of (Î¾t)tâ¥0 until the first branch time, 1N . At this time, a fair coin is flippedfor each individual: if it comes up heads (with probability 12) the individual dies. If it comesup tails, the individual branches into two offspring at the same location. The branchingevent occurs independently for each individual and produces a new generation of individualswhich then evolve in space as independent copies of (Î¾t)tâ¥0 until the next branching time,2N . The reproduction procedure is then repeated, and the process continues to evolve in thesame fashion.By assigning a mass of 1N to each individual, we can view the state at time t > 0 asa mass distribution. In particular, if there are Nt surviving individuals at time t withlocations xi for i = 1, . . .Nt, we consider the empirical measureX(N)t =1NNtâi=1Î´xi ,where δx denotes the Dirac mass at x â Rd. (A proper analysis of this process requiresa more sophisticated labelling system for the individuals, but we do not require this forour description here.) By fixing the particle mass at 1N , the expected total mass of thepopulation remains constant for allN . The acceleration of the branching rate (i.e. branchingevents at every multiple of 1N ) ensures that the variance of the total mass at time t remainsconstant in N .The larger N is, the greater the size of the population and the smaller the contributionof a particular individual to the associated mass distribution. By shrinking the intervalbetween branching times we are effectively accelerating time. Therefore for large values ofN we are observing a large population over long time-scales, normalized to have constantexpected total mass.A common theme in probability theory is to approximate the large scale behaviour ofa discrete random system with a scaling limit, which is a continuum model that arises asthe distributional limit of the rescaled discrete models. The most famous example is thecentral limit theorem. The class of processes we study in this thesis, superprocesses, are thescaling limits of branching particle systems such as the one described above. In additionto their connection with numerous discrete models (including more complex models thanthe one we have described), superprocesses are objects of independent interest. They alsoare deeply connected with certain families of non-linear elliptic and parabolic differentialequations.21.2 Continuous state branching and superprocessesThe goal of this section is to introduce and define the Dawson-Watanabe superprocess,our central object of study. We introduce superprocesses in the broader context of scalinglimits for branching processes. This means that before considering spatial branching modelsand superprocesses we dedicate some discussion to non-spatial branching processes. Theidea behind our presentation is that while most readers will be familiar with the theory ofMarkov processes in Rd (i.e. the processes which describe the spatial motion in our models),some may be unfamiliar with continuous-state branching.1.2.1 Warm up: a limit theorem for branching processesWe begin with some classical limit theorems for branching processes. Let µ be a probabilitydistribution on N with probability mass function (pmf) µ({n}) = µn. For a functionf : N â R, we denote the expectation by µ(f) = âân=0 f(n)µn. We make the followingassumptions on µ:â¢ µ(Id) =âân=0 nµn = 1 (Criticality).â¢ âân=0 n2µn < +â (Finite second moment).â¢ µ1 < 1 (Non-degeneracy).Furthermore we suppose that µ has variance γ > 0.A Galton-Watson branching process, or simply a branching process, with offspring dis-tribution µ is a Markov chain with state space N with the following time-homogeneoustransition probabilities:P (Zn+1 = k |Zn = m) = (µ â µ â · · · â µ︸ ︷︷ ︸m)(k).The state of the chain Zn denotes the number of individuals in the population at timen. Conditioned on Zn = m, the distribution of Zn+1 is the m-fold convolution of theoffspring distribution µ. Of course, this corresponds to the intuitive description that foreach individual in the population at time n, we independently sample a number of offspringfrom µ, and the size of the population at time n+1 is the total number of offspring producedby generation n.The classical analysis of branching processes is often done using generating functions.Let X be a random variable with distribution µ. We denote the probability generatingfunction of X by f(s) = E(sX). It follows from an elementary conditioning argument thatfn(s) = f(fnâ1(s)) (1.2.1)3where fn(s) = E(sZn) is the (probability) generating function of Zn. In particular, if Z0 = 1deterministically, fn(s) is equal to the n-fold composition of f with itself. This recursion isclassically used to determine the extinction probability (that is, the probability that Zn = 0for some n) of the branching process, which can be shown to be equal to the smallest positivefixed point of f , i.e. the smallest s0 > 0 such that f(s0) = s0. We have assumed that ourbranching process is critical, in which case the extinction probability is one. More detailedinformation, such as the asymptotics of the survival probability, can also be obtained usingthis method. Suppose that Z0 = 1. From (1.2.1), it follows thatE(Zn) = µ(1)nZ0 = 1 (1.2.2)for all n â¥ 1. We can also compute the second moment of Zn, which yieldsVar(Zn) = nγ. (1.2.3)By the second moment method, we deduce the following: when Z0 = 1,P (Zn > 0) â¥ E(Zn)2E(Z2n)â¥ 12nγ + 12â 1nγ. (1.2.4)This estimate is quite good and only differs from the actual asymptotics by a factor of 2.The exact asymptotics of the survival probability were originally established by Kolmogorov[44].Theorem 1.2.1. If Z0 = 1, thenlimnâânP (Zn > 0) =2γ.This result can be proved using a more refined analysis of generating functions. See, forexample, Theorem II.1.1 of [83].We now present Fellerâs Theorem, which describes the diffusion approximation, or scalinglimit, of the branching process. The most famous diffusion approximation in probabilitytheory is undoubtedly Donskerâs Theorem, which states that the scaling limit (with diffusivespace-time rescaling) of a random walk with finite variance step distribution is Brownianmotion. Fellerâs Theorem is an analogous statement which characterizes the scaling limit ofbranching processes whose offspring distributions have finite second moments as the solutionto a stochastic differential equation.Let a > 0 and suppose µ is an offspring distribution satisfying the assumptions statedat the beginning of the section. We define a sequence of stochastic processes (Z(N)t )tâ¥0 asfollows. Let ZË(N)n be a branching process with offspring distribution µ and ZË0 = âaNâ. For4t > 0, we define Z(N)t byZ(N)t =1NZË(N)âtNâ. (1.2.5)We will write R+ to denote the non-negative real axis [0,â). For a Polish space E, denoteby D(R+, E) the space of ca`dla`g (right-continuous with left limits) paths in E with timeindex set R+, equipped with the Skorokhod topology. (For a general introduction to thisspace and topology, see Chapter 3 of Ethier and Kurtz [23].) The following is due to Feller[26].Theorem 1.2.2. (Z(N)t )tâ¥0 â (Zt)tâ¥0 weakly on D(R+,R+), where Zt is the unique-in-lawsolution toZt = a+â« t0âÎ³ZsdBs (1.2.6)The solution to the stochastic differential equation (SDE) (1.2.6) is called Fellerâs branch-ing diffusion. Solutions to the SDE are not only unique-in-law, but satisfy pathwise unique-ness by the Yamada-Watanabe theorem (see Theorem V.40 of [86]). The martingale problemformulation of (1.2.6) is a non-spatial version of the martingale problems which can be usedto define some superprocesses; however, here we use an approach based on the Laplacefunctional.1.2.2 Continuous state branching with general branching mechanismsIn this section we give a general limit theorem for branching processes and characterizationof their limits, continuous state branching processes (CSBPs). The state space of a CSBPis R+ rather than N, and the state is understood as the total mass of a population ratherthan the number of individuals. The continuous-state branching behaviour we discuss isclosely related to the theory of infinitely divisible distributions and Le´vy processes.The first work considering continuous state branching in some generality was by JiËrinain 1958 [40]. Other early work includes articles of Lamperti [49, 50, 51] and Silverstein[87], some of which we discuss below. The construction we adopt is based on that used byDawson [13] to construct superprocesses, which we employ again in the next section whenwe do the same. Here we restrict our attention to branching processes with a single type,but we note that continuous state branching models with multiple types have been studiedextensively (early work includes [88] and [93]) and remain an area of active interest (somerecent works include [47, 48]).Consider a continuous-time branching process whose offspring distribution has probabil-ity mass function (pk)kâ¥0. The probability generating function of the offspring distributionis theng(z) =ââk=0zkpk. (1.2.7)5In the continuous-time setting, individuals no longer reproduce simultaneously at eachtime step to produce the next generation; rather, we fix κ > 0 and attach an independentExponential(κ) clock to each individual. When the clock sounds, the individual is replacedwith a random number of offspring sampled from the offspring distribution and to eachoffspring is attached an independent Exponential(κ) clock.The process equal to the number of individuals in the population is a strong Markovprocess with state space N. We will denote it by (Zt)tâ¥0, and for the law and expectationof (Zt)tâ¥0 with Z0 = m â¥ 1, offspring generating function g(z) and birth rate κ, we willwrite P g,κm and Eg,κm respectively.A critical tool for analysing many processes with branching behaviour is the Laplacefunctional, i.e. the quantityL(m,λ, t) = Eg,κm (exp(âÎ»Zt)) (1.2.8)for λ > 0, m â N and t > 0. We first remark that this quantity satisfiesL(m,λ, t) = L(1, λ, t)m. (1.2.9)This is known as the branching property. It reflects, and is an immediate consequence of,the fact that the lineages extending from separate individuals in the branching process areindependent. It implies that it is enough to analyse L(1, λ, t), and so to denote this quantitywe will simply write L(λ, t). By conditioning on the time and size of the first branchingevent, it is elementary to show thatL(λ, t) = exp(â(Î»+ κt)) +â« t0g(L(λ, tâ s))ÎºeâÎºsds. (1.2.10)The fact that the Laplace functional of a branching process is related to the solution of anevolution equation is essential. Indeed, an analogous relationship holds for superprocessesand the methodology we adopt is centred around it. We will come to this in the next section.The Laplace functional fully characterizes the distribution of Zt (and more generally of anyR+-valued random variable) so it is a robust framework from which to study these processesand their distributional limits.We consider a scaling regime in which Z0 = O(N) and the mass of each individual isNâ1. In addition to large population size, we wish to study the process over large time-scaleswhich we achieve by using a branching rate κN with limNââ κN =â.A random variable is called infinitely divisible if for every n â¥ 1, it can be written asa sum of n iid random variables. If Z0 = N , by the branching property we can write Ztas a sum of N independent branching processes starting with one individual. We should6therefore expect that, once renormalized to ensure tightness, the scaling limit of Zt for anyfixed t > 0 should be infinitely divisible. Indeed, this is the case. Infinite divisibility of themarginal distributions and the branching property are sufficient to characterize the scalinglimits of branching processes, continuous state branching processes.We now recall the Le´vy-Khintchine formula for infinitely divisible random variables.(For a proof, see Section 9.5 of [4].)Theorem 1.2.3. A random variable X is infinitely divisible if and only if its characteristicfunction (Fourier transform) satisfiesE(exp(iuX)) = exp(âÎ¨(u)), (1.2.11)where the characteristic exponent Ψ is of the formΨ(u) = ibu+γ2u2 +â« âââ(1â eiur + iur1|r|â¤1)Î½(dr), (1.2.12)where b â R, γ â¥ 0, ν is a measure on R with ν({0}) = 0 and â« (1 â§ |r|2)ν(dr) <â.The characteristic function is often used because it is well-defined regardless of thedistributional properties of the variable. However, the variables and processes we considerare spectrally positive, which means that the Le´vy measure ν is supported on (0,â). Thiscorresponds to a process with no negative jumps. In this case the Laplace transform is theextension of the Fourier transform from the imaginary axis to the negative real numbers.For a spectrally positive infinitely divisible random variable X, we haveE(exp(âuX)) = exp(âÏ(u)), (1.2.13)where the Laplace exponent Ï(u) is of the formÏ(u) = muâ γ2u2 +â« â0(1â eâur â ur1|r|â¤1)Î½(dr), (1.2.14)with m â R, γ â¥ 0 and ν a measure on (0,â) satisfying â«â0 (1 â§ r2)ν(dr) < â. In thecontext of CSBPs, the Laplace exponent Ï(u) is often called a branching mechanism. If,in addition, we haveâ«â1 rν(dr) < â, we say that Ï has finite mean. By redefining theconstant m we obtain that the general form of a branching mechanism with finite mean isÏ(u) = muâ γ2u2 +â« â0(1â eâur â ur) ν(dr), (1.2.15)withâ«â0 râ§r2Î½(dr) <â. As we discuss later, the value of Ïâ²(0+) gives the exponential rate7of growth of the mean of the CSBP. If Ï has representation (1.2.15), then Ïâ²(0+) = m andthe branching mechanism is called sub-critical, critical and super-critical corresponding tothe cases m < 0, m = 0 and m > 0.Our primary goal is to construct a broad class of CSBPs and thereby characterize theclass of Markov processes which arise as scaling limits of branching processes. The limittheorem we state is general enough to construct every CSBP whose branching mechanism iscritical (and therefore has finite mean). In order to construct these processes, we first choosethe branching mechanism Ï(u) and define a sequence of offspring distribution generatingfunctions gN and branching rates κN , and show that the sequence of branching processeswith parameters (gN , κN ) converges in distribution to a limit process with branching mech-anism Ï. In doing so, we restrict the offspring distributions under consideration. However,the method we adopt is sufficient to build all CSBPs in the class we consider. If a sequenceof critical branching processes associated to a generic sequence (gN , κN ) is tight, then itsdistributional (possibly subsequential) limit will be one of the CSBPs constructed in thesequel, and in this sense the sequence (gN , κN ) is a perturbation of one of the sequences weconsider.Let Ï(u) be a critical branching mechanism (i.e. Ï is of form (1.2.15) with m = 0). Inour choice of the scaling regime we follow Section 4 of Dawson [13]. For N â N, letκN = âÏâ²(N), (1.2.16)andgN (z) = Îºâ1N [κNz âNâ1Ï (N(1â z))] (1.2.17)For a > 0 let MN be a Poisson(aN) random variable. We wish to establish the convergenceof the sequence of processes (Nâ1Z(N)t )tâ¥0, where Z(N)t has law PgN ,κNMN. The following isthe special case of Theorem 4.4.1 of [13] when there is no spatial motion.Theorem 1.2.4. Let Ï(u) be a critical branching mechanism, and for N â¥ 1 and a â¥ 0, letκN , gN , MN be as above. Then the process (Nâ1Z(N)t )tâ¥0, where (Z(N)t )tâ¥0 has law PgN ,κNMN,converges in distribution to an R+-valued strong Markov process (Yt)tâ¥0 with Y0 = a, whereYt is the strong Markov process with the following Laplace functional: for x â¥ 0 and λ > 0,EYx (exp(âÎ»Yt)) = exp(âxuÎ»(t)), (1.2.18)where uλ(t) is the unique solution touâ² = Ï(u), u(0) = λ. (1.2.19)8The process (Yt)tâ¥0 in the limit above is called the Ï-continuous state branching pro-cess, or Ï-CSBP. Lamperti was the first to establish CSBPs as the distributional limitsof branching processes [50, 51]. The limit theorem we have stated is restricted to criticalbranching mechanisms; however, for all Ï of the form (1.2.14), the Ï-CSBP exists and itsLaplace functional is characterized by (1.2.18) and (1.2.19).Let Ï be a finite mean branching mechanism with representation (1.2.15) and considerthe Ï-CSBP Yt. The state Yt represents the total mass of a population. As can be seendirectly from (1.2.18), Yt satisfies the branching property: for t, x, y, λ > 0,EYx+y(exp(âÎ»Yt)) = exp(â(x+ y)uλ(t)) = EYx (exp(âÎ»Yt))EYy (exp(âÎ»Yt)). (1.2.20)In particular, for every N â¥ 1 we can writeEYx (exp(âÎ»Yt)) = (EYx/N (exp(âÎ»Yt)))N , (1.2.21)and hence Yt is infinitely divisible.The branching mechanism Ï encodes the properties of the Ï-CSBP. For example, aCSBP may demonstrate finite time extinction. Consider the extinction time Ï0 = inf{t >0 : Yt = 0}. It was shown by Grey [31] that (for any a > 0), P Ya (Ï0 < â) = 1 if and onlyif, for sufficiently large θ > 0, Ï(u) < 0 for u â¥ θ andâ« âÎ¸1|Ï(u)|du <â.Blow-up behaviour is also governed by Ï [31]. However, this does not occur in the finitemean setting to which we have restricted.We recall our earlier definition that Ï (and the Ï-CSBP) is sub-critical, critical, andsuper-critical respectively when m < 0, m = 0 and m > 0. It can be shown via anelementary calculation using (1.2.18) and (1.2.19) that EY1 (Yt) = emt. Another result ofGrey [31] states that P Ya (Ï0 =â) > 0 if and only if Yt is super-critical. This is analogous tothe situation for discrete branching processes, which survive forever with positive probabilityif and only if they are super-critical.The Le´vy-Khintchine formula characterizing infinitely divisible distributions is closelyrelated to the theory of Le´vy processes. A Le´vy process is an Rd-valued stochastic processwith stationary, independent increments. An immediate consequence is that its distributionat time t > 0, say Wt, is infinitely divisible. Because the increments are stationary andindependent, it follows that the distribution of a Le´vy process is completely characterizedby its characteristic exponent, which satisfies the Le´vy-Khintchine formula. The followingcharacterization of CSBPs as time-changed Le´vy processes is due to Lamperti [49].9Theorem 1.2.5. Let Ï be a branching mechanism, Wt be the Le´vy process with Laplaceexponent Ï, and WËt be the process Wt stopped at 0. For t > 0 we define U(t) = inf{Ï :â« Ï0 WËâ1s ds = t}. Then (WËU(t))tâ¥0 is a Ï-CSBP. Moreover, all continuous state branchingprocesses can be obtained in this way from a Le´vy process with no negative jumps.We will primarily be interested in the two following critical finite mean branching mech-anisms: Ï(u) = âÎ³2u2 with γ > 0 and Ï(u) = âu1+Î², for 0 < β < 1. We note that for bothcases we have Ïâ²(0+) = 0 andâ«â1 |Ï|â1 < â, so both are critical and go extinct in finitetime almost surely by Greyâs result mentioned above.Now consider the case Ï(u) = âÎ³2u2. The Ï-CSBP is Fellerâs branching diffusion withbranching rate γ, which we have seen in the previous section. We saw that this processarises as the scaling limit of any branching process whose offspring distribution has finitevariance, and that it is the solution to the stochastic differential equation (1.2.6). It is thusthe Gaussian CSBP (while it is not a Gaussian process per se, it is driven by Brownianmotion). In view of Theorem 1.2.5, Fellerâs branching diffusion is a time-changed Brownianmotion stopped at 0.For Ï(u) = âÎ³2u2, the branching is strictly binary. Although a CSBP describes anevolving population mass, and a priori does not have âindividuals,â it is possible to studyits genealogical structure and indeed recover its entire genealogical tree. In the case ofÏ(u) = âÎ³2u2, the genealogical tree is Aldousâ Brownian Continuum Random Tree, whichis strictly binary almost surely. We discuss the genealogies of continuous state branchingmodels at the end of Section 1.2.3.Let us recall momentarily the general form of a finite mean branching mechanism:Ï(u) = âÎ³2u2 +mu+â« â0(1â eâur â ur) ν(dr).So Ï(u) is a sum of three terms. The first is the one we have just considered in isolation andcorresponds to binary branching. The second term mu is a drift term; the sign of m governswhether the process is critical, sub-critical or super-critical, and hence the asymptotics ofthe mean. The last term allows the most freedom and is determined by the Le´vy measure,or jump measure, ν. It follows that, much like Brownian motion (with drift) is the onlyLe´vy process with continuous paths, Fellerâs branching diffusion (possibly with drift) is theonly CSBP with continuous paths. The rest have discontinuous birth events, i.e. positivejumps.For 0 < β < 1, one can obtain Ï(u) = âu1+Î² by taking m = γ = 0 and defining ν byν(dr) = cÎ²râ2âÎ²dr for r > 0. Let us denote the associated CSBP by Yt. Since the onlynon-zero term in Ï is ν, Yt grows via upward jumps. In the Nth branching process in the10approximation scheme we described above, the branching rate isκN = (1 + β)Nβ .This implies that the appropriate time-scale on which to observe such branching processesis different than in the finite variance case, which we saw in Theorem 1.2.2. We also havethat for each N â¥ 1,gN (z) = z +11 + Î²(1â z)1+β .This is the generating function of a heavy-tailed random variable in the domain of attractionof a positive stable random variable of index 1 + β, whose pth moment is infinite for allp â¥ 1 + β. (For a complete description of the stable laws and their associated central limittheorems, see Chapter 9 of Breiman [4].)The Le´vy process associated to Yt (c.f. Theorem 1.2.5) is a spectrally positive (1 + β)-stable process. For this reason, we will call Yt the (1 + β)-stable branching CSBP. Oneconsequence of the stable (heavy) tails of the jump measure of Yt is thatEYa (Y1+θt ) = +âfor θ â¥ β, which is expected in view of the offspring distribution of the associated branchingprocesses. If θ â (0, β), the moment is finite. As is common throughout probability theory,the analysis of processes and variables with infinite second moments often requires differentmethods than those used to study processes with finite second moments.1.2.3 Spatial branching particle systems, superprocesses, and their dualsWe now describe our main object of study: superprocesses. Superprocesses are to spatialbranching particle systems as CSBPs are to branching processes. We will first define thesuperprocess associated to a given spatial motion and branching mechanism, then brieflydescribe a class of branching particle systems which converge in distribution to the super-process.The state of a superprocess is a spatial mass distribution, so the natural state space isthe space of measures. Let MF (Rd) denote the space of finite measures on Rd, whichwe endow with the topology of weak convergence. Superprocess paths then reside inD([0,â),MF (Rd)).We now describe the spatial motion underlying the superprocess. Let ξt be a time-homogeneous Borel strong Markov process on Rd with transition semigroup (St)tâ¥0. Wewill write Bb to denote the space of bounded Borel functions on Rd, and B+b to denote those11that are non-negative. For Ï â B+b , we have(StÏ)(x) = EÎ¾x(Ï(Î¾t)).Generally, St is a strongly continuous contraction semigroup on a Banach space (e.g. Lpfor 1 â¤ p <â). The infinitesimal generator of ξt, which we denote A, is a closed operatorwith dense domain D(A) in that Banach space such that for all Ï â D(A),limtâ0+StÏâ Ït= AÏ.Moreover,ddtStÏ = AStÏ = StAÏ.We now define the superprocess associated to A and Ï, where A is as above and Ïis as in (1.2.14). The definition we give specifies a measure-valued Markov process via itsMarkov transition kernel. For µ â MF (Rd) and Ï â B+b , we write µ(Ï) =â«Ï(x)µ(dx).Definition 1.2.6. The (A, Ï)-superprocess is the MF (Rd)-valued Markov process Xt with(time-homogeneous) Laplace functional characterized as follows: for t > 0, X0 â MF (Rd)and Ï â B+b ,EXX0(exp(âXt(Ï)) = exp(âX0(ut)), (1.2.22)where ut(x) is the unique solution (with initial condition u0(x) = Ï(x)) tout(x) = (StÏ)(x) +â« t0Stâs(Ï(us))(x) ds (1.2.23)for (t, x) â (0,â) Ã Rd.For Ï â B+b , we will write uÏt (x) and uÏ(t, x) interchangeably to denote the uniquesolution to (1.2.23). The evolution equation (1.2.23) is a mild form of the partial differentialequation (PDE) ï£±ï£²ï£³ââtu(t, x) = Au(t, x) + Ï(u(t, x))u(0, x) = Ï(x)(1.2.24)for (t, x) â (0,â) Ã Rd. When a unique solution to (1.2.23) exists, generally it is alsothe unique solution to (1.2.24). Probabilists tend to work with the mild form to avoidtechnicalities such as regularity, but the differential form is useful for deriving heuristics.If u0(x) = Ï(x) â¡ λ > 0, we have Au(t, x) â¡ 0 for all (t, x) â (0,â) Ã Rd, and itfollows that u(t, x) is equal to the (constant in x) solution to (1.2.19). We therefore seethat the total mass process Xt(1) is a Ï-CSBP. All non-spatial properties such as criticality,12survival/extinction probabilities, and blow-up are completely determined by Ï.The (A, Ï)-superprocess has the branching property. Suppose that µ, ν â MF (Rd), andconsider Xt with X0 = µ+ ν. It then follows from (1.2.22) that for all Ï â B+b we haveEXµ+Î½(exp(âXt(Ï)) = EXµ (exp(âXt(Ï))EXÎ½ (exp(âXt(Ï)). (1.2.25)An immediate consequence of the branching property is that Xt is an infinitely divisiblerandom measure. (Intuitively, a random measure X is infinitely divisible if X(Ï) is aninfinitely divisible random variable for all suitable test functions Ï.)On an heuristic level, the (A, Ï)-superprocess is a Ï-CSBP which lives in Rd, in whichthe âindividualsâ (or âparticlesâ) move in space like independent copies of the Markovprocess generated by A. The difficulty with this description is that a priori there are noparticles in the continuous state space setting. One can envision the situation as infinitelymany particles of infinitesimal mass which branch with infinite branching rate. In fact,this heuristic can be made rigorous using non-standard analysis. The use of non-standardmodels to study superprocesses was pioneered by Perkins in [78, 79, 80] and was also usedin [14] and [90]. Other methods which recover the individual-based approach are discussedlater in this section.The first limit theorem establishing a superprocess as the limit of a branching particlesystem was from Watanabe [92] in 1968. (In fact, Watanabe proved the convergence of finitedimensional distributions; an early proof of tightness is due to Roelly-Coppoletta [85].) Wewill give a relatively general limit theorem which is due to Dawson [13], who constructed abroad array of superprocesses using duality and the Laplace functional. We note in passingthat the class of superprocesses under consideration are those that at one point were calledDawson-Watanabe superprocesses.Let Ï be a critical branching mechanism. For N â¥ 1 we define a continuous timebranching process with branching rate and offspring generating function κN and gN as in(1.2.16) and (1.2.17), only now the population lives in space. Fix µ â MF (Rd) and let NNbe a Poisson point process on Rd with intensity measure Nµ. The population begins withone individual at each point in NN . Each individual carries an Exponential(κN ) clock andmoves like an independent copy of ξt until the clock sounds. At this time, the individualis replaced with a random number of offspring sampled according to gN which appear atthe parentâs location. They are all given clocks and evolve spatially like independent copiesof ξt. We denote the empirical measure of the process at time t â¥ 0, which we obtain byplacing a point mass at the location of each living individual, by X(N)t . The following isstated in [13] as Theorem 4.4.1.Theorem 1.2.7. Let A generate a Borel strong Markov process, Ï be a critical branching13mechanism, and (κN , gN ) be given by (1.2.16) and (1.2.17) for N â¥ 1. Then the process(Nâ1X(N)t )tâ¥0 with initial configuration NN converges in distribution (on D([0,â),MF (Rd)))to the (A, Ï)-superprocess with X0 = µ.The definition we have given is not the only way to characterize superprocesses, but itis one of the most general. Here we mention a few other approaches. One possibility is todefine a superprocess as a Markov process directly by specifying its generator. The stateis a measure, so the generator acts on functions of measures. See, for example, Section 4.2of [22] or the introduction of [45]. The theory of martingale problems is also well-suited todefining and studying superprocesses. This is the approach used by Perkins in [83]. See alsoChapter 9 of Ethier and Kurtz [23]. The martingale problem formulation is effective whenproving that the density of a superprocess (when it exists) is the solution to a stochasticpartial differential equation (SPDE); see [45] and [70].It is often useful and interesting to consider not only the population size in a branchingprocess but its genealogical structure. This is conceptually simple in the discrete case.One simply considers the entire family tree and keeps track of the ancestral data of eachindividual. There are no major mathematical difficulties in making this rigorous.It is not as obvious how to do so, but the genealogical structure of continuous statebranching models can be described. We briefly mention a few approaches. The first wediscuss is the historical process of Dawson and Perkins [15]. The historical process Htis an inhomogeneous Markov process with state space (at time t > 0) MF (D([0, t],Rd)),the finite measures on ca`dla`g paths in Rd. The paths in Ht correspond to lineages of thesuperprocess which survive until time t. The historical process therefore encodes all thegenetic (historical) information of Xt, and one can recover Xt by projecting Ht onto Rd byidentifying paths in D([0, t],Rd) with their endpoint.The Brownian Continuum Random Tree (BCRT) was defined by Aldous in [1]. It isthe distributional limit, in an appropriate space, of many families of discrete random trees,including the ancestral trees of branching processes which converge to Fellerâs branchingdiffusion (the Ï-CSBP with Ï = â12u2). In particular, the genealogical tree in binarybranching CSBPs and superprocesses is the BCRT. In [2], Aldous proposed realizing thebinary branching superprocess as a sort of tree-indexed Brownian motion relative to theBCRT as a means of describing the process in a way that incorporates genealogical informa-tion. This was ultimately achieved by Le Gall with the introduction of the Brownian snakein a series of works in the 1990s [52, 53]. Later work of Le Gall with Le Jan [58, 59] andDuquesne [18] constructed general Le´vy trees, a family of continuum random trees general-izing the BCRT, and the associated Le´vy snake, and used it to give a complete genealogicaldescription of superprocesses with general branching mechanisms.Both the historical process and the Brownian snake give a means of realizing the su-14perprocess in a way which includes genealogical data. The inclusion of ancestral pathsrecovers, in a sense, the notion that the mass in Xt corresponds to the spatial distributionof massive individuals, although of course the number of individuals is infinite and theirmass infinitesimal. Another method for âkeeping track of the individualsâ in continuousstate branching models is the so-called âlookdownâ model. In the lookdown approach, eachindividual has an associated âlevel,â or index, and individuals are ordered correspondingto the longevity of their ancestral lines. (Imposing this without looking into the futurerequires the use of exchangeability.) Donnelly and Kurtz introduced this method in [16]and applied it to superprocesses in [17]. The latter work included a construction of thehistorical process using a lookdown model. Many types of processes, including branchingparticle systems and genetic models, can be realized using a lookdown model. The variantof the lookdown approach developed by Kurtz and Rodriguez in [46] can also be applied tosuperprocesses.This thesis is concerned with critical superprocesses when X0 (and hence Xt) has fi-nite mass. There is a literature concerned exclusively with the super-critical regime, of-ten focused on asymptotic behaviour, but we do not discuss this here. In some scenariossuper-critical superprocesses can be obtained from their critical counterparts via a Girsanovtransformation for superprocesses; see Theorem 7.7.2 of [13]. Asymptotic and ergodic be-haviour has also been considered in the regime with infinite mass and critical branching.One example is the persistence/extinction dichotomy proved by Dawson [11].1.3 Two superprocesses and their propertiesWe now focus our attention on the two superprocesses which are studied in this thesis:super-Brownian motion and the (α, β)-superprocess. We define these processes and reviewsome of their fundamental properties.1.3.1 Super-Brownian motionSuper-Brownian motion is the (A, Ï)-superprocess with A = 12â and Ï(u) = âÎ³2u2, withγ > 0. Hereafter we fix γ = 1. We recognize this branching mechanism as correspondingto Fellerâs branching diffusion, and so the branching is binary and the total mass processis continuous. The individuals move in space like Brownian motion, since 12â generatesBrownian motion in Rd. Because binary branching and Brownian motion are of considerableinterest, super-Brownian motion is of central importance in the study of superprocesses. Itis also the best studied.We will now specialize our notation so thatXt denotes a super-Brownian motion (insteadof a general superprocess), and for its law and expectation with initial value X0 we will write15PXX0 and EXX0, respectively. In this case, (1.2.22) and (1.2.23) take the following form: forall Ï â B+b ,EXX0(exp(âXt(Ï))) = exp(âX0(V Ït )), (1.3.1)where V Ït (x) is the unique solution toVt(x) = StÏ(x)â 12â« t0Stâs(V 2s ) ds, (1.3.2)where St denotes the transition semigroup of Brownian motion (the heat semigroup). Wewill sometimes use the alternate notation V Ï(t, x) to denote the unique solution to (1.3.2).The evolution equation (1.3.2) is a mild form of the PDEââtV (t, x) =12âV (t, x)â 12V 2(t, x). (1.3.3)1.3.2 The (α, β)-superprocessFor 0 < α â¤ 2 and 0 < β â¤ 1, the (α, β)-superprocess is the superprocess with Ï(u) =âu1+Î² and A = â(ââ)Î±2 , the fractional Laplacian. We note that α = 2 and β = 1 recoverssuper-Brownian motion, which we have discussed separately. For our purposes we willassume that α < 2 but allow for β = 1, which corresponds to binary branching.We recall from Section 1.2.2 that the branching mechanism Ï(u) = âu1+Î² with β < 1induces stable branching, which has two important properties: self-similarity and heavytails. Recall as well that branching of this type has discontinuous birth events at which apositive mass is added to the total population. In the spatial setting, this means that birthevents correspond to the addition of a point mass cδx to the state, with mass c > 0 and atlocation x â Rd, and that the set of jump times is dense in [0, Ï0), where Ï0 is the extinctiontime (see Section 6.2.2 of [13]).For 0 < α < 2, the fractional Laplacian â(ââ)Î±2 is the generator of the isotropicα-stable process, which we will often simply call the α-stable process. This process is aself-similar, pure jump Le´vy process. It has discontinuous paths and its transition densityhas power-law decay. We will use the notation âÎ± = â(ââ)Î±2 .We will denote the (α, β)-process by Xt. With initial measure X0 â MF (Rd), we willwrite PXX0 and EXX0 to denote its law and expectation. The Laplace functional is thenexpressed as follows: for Ï â B+b ,EXX0(exp(âXt(Ï))) = exp(âX0(uÏt )), (1.3.4)16where uÏt (x) = uÏ(t, x) is the unique solution tout(x) = Sαt Ï(x)ââ« t0SÎ±tâs(u1+Î²s )(x) ds. (1.3.5)Here Sαt denotes the semigroup of the α-stable process (the heat semigroup of âÎ±). Theassociated PDE is thenââtut(x) = âÎ±ut(x)â u1+βt (x). (1.3.6)1.3.3 Super-Brownian motion with (1 + β)-stable branchingIn the above we have excluded the natural case of A = 12â and Ï(u) = âu1+Î² with β < 1.This superprocess is called super-Brownian motion with (1 + β)-stable branching. We donot emphasize this process because none of the results in this thesis involve it.1.3.4 SupportsBecause superprocesses describe randomly evolving spatial mass distributions, their sup-ports are of great interest. By support we mean the closed support and will write S(µ) todenote the support of µ â MF (Rd). In this section we review some classical results con-cerning the supports of super-Brownian motion and the (α, β)-superprocess. (We will alsobriefly discuss Borel supports.) In some cases, this gives us the opportunity to showcasethe methodology we adopt to prove some of our own results.For both Xt and Xt, there is a parameter regime in which the measure is absolutelycontinuous with respect to Lebesgue measure and has a density. In particular, Xt is abso-lutely continuous if and only if d = 1, and Xt is absolutely continuous if and only if d < αβ[27]. The new results of this thesis are about the densities and so exclusively hold in theabsolutely continuous regime. We discuss the densities in detail in the next section. Inthe sequel, we discuss a variety of properties of the supports of Xt and Xt, some of whichare specific to the non-absolutely continuous regime and are therefore not germane to theresults of this thesis. We include them nonetheless in the interest of situating our resultswithin a broader context.We begin with super-Brownian motion. For a Borel set A â Rd, we say that XÂ· evercharges A if Xt(A) > 0 for some t > 0. We will write B(x,R) to denote the open ball ofradius R > 0 centered at x, and B(x,R) to denote its closure. The following is due to Iscoe[38].Theorem 1.3.1. Let X0 â MF (Rd) such that S(X0) â B(0, R0), for 0 < R0 < R. ThenPXX0(XÂ· ever charges B(0, R)c) = 1â exp(âRâ2X0(v(Râ1·))),17where v : B(0, 1)â R+ is the unique, positive, radial solution toï£±ï£²ï£³âv(x) = v2(x), |x| < 1v(x)â +â as |x| â 1.(1.3.7)Corollary 1.3.2. If X0 â MF (Rd) has compact support, âªtâ¥0S(Xt) is compact PXX0-almostsurely.Thus S(Xt) is compact1 and the probability that XÂ· ever charges a set is expressedin terms of the solution to a singular boundary problem for a non-linear elliptic equation.To see how this arises, let us first consider the state at a fixed time t > 0. SupposeS(X0) â B(0, R0) and 0 â¤ R0 < R. From (1.3.1), for λ > 0 we haveEXX0(exp(âÎ»Xt(B(0, R)c)) = exp(âX0(V λ,Rt )), (1.3.8)where V λ,Rt is the solution to (1.3.3) with V0 = λ1B(0,R)c . Taking Î»ââ yieldsPXX0(Xt(B(0, R)c) = 0) = exp(âX0(Vâ,Rt )), (1.3.9)where Vâ,Rt (x) = limÎ»ââ Vλ,Rt (x). Informally, Vâ,Rt is the solution to (1.3.3) with V0 =â1B(0,R)c . The probability that Xt charges B(0, R)c at fixed t > 0 is expressed in terms ofthe solution to a singular initial value problem to the dual equation.The approach used to prove Theorem 1.3.1 is similar. It uses what is called the occupa-tion time process of Xt, namelyXt(Ï) =â« t0Xs(Ï) ds. (1.3.10)The event that XÂ· ever charges A â Rd is equivalent to X(A) > 0, where X is the integratedsuper-Brownian motion X = limtââXt. The Laplace functional of Xt is connected to aninhomogeneous version of the regular dual equation (see [38]). The proof of Theorem 1.3.11The compact support property is sometimes called finite speed propagation. From the perspective ofBrownian motion, it is not surprising that S(Xt) is compact; Brownian paths are continuous, and everypoint in S(Xt) is the endpoint of a Brownian path. Of course, there are infinitely many such paths, so apriori this does guarantee that the collection of their endpoints is bounded. However, the strong dependencestructure between them, namely the genealogical structure, causes S(Xt) to be compact.Another perspective is as follows. If we consider super-Brownian motion with branching rate γ > 0,(i.e. with branching mechanism Ï(u) = â γ2u2) the limit of the dynamics as γ â 0 is simply heat flow,which famously has infinite-speed propagation. But for any γ > 0, the support is compact (although, onaverage, its diameter is larger for smaller values of γ). One can view this as the noise, that is the randomnessassociated to branching, slowing down the propagation of the mass distribution. An example of a similarphenomenon is a celebrated result showing that the speed of stochastic travelling waves solving a stochasticFKPP equation is slower than the classical travelling wave solutions of the FKPP equation [68].18considers a family of solutions to the inhomogeneous equation and ultimately establishestheir convergence as t â â to the solution of (1.3.7). There is also a direct connectionbetween super-Brownian motion and the elliptic equation âv = v2 due to Le Gall [54] whichuses the Brownian snake and exit measures to represent these solutions probabilistically.This development occurred after the original proof of the compact support property, but itgives a direct proof.Besides XÂ· ever charging the complement of a large ball, this methodology can beadapted to other events. In the same paper, Iscoe considered the probability that XÂ·ever charged B(x, Ç«) when X0 = δ0 and determined its asymptotics for large |x| and smallǫ. In [14], Dawson, Iscoe and Perkins considered similar events but with Xt rather than X.They gave tight bounds on the probability that Xt charges a given set before some timet0 > 0 which we use in a proof in this thesis.The support process of super-Brownian motion is right continuous with a one-sidedmodulus of continuity. For A â Rd and δ > 0, let Aδ = {x â Rd : d(x,A) â¤ δ}, whered(x,A) = infyâA |x â y|. For r > 0, let h(r) =âr log(râ1) â¨ 1. The following is due toDawson, Iscoe and Perkins [14].Theorem 1.3.3. (a) For every c > 2, a.s. there exists δ(c) > 0 such that if t, s â¥ 0 with0 < tâ s â¤ δ(c), then S(Xt) â S(Xs)ch(tâs).(b) The map tâ S(Xt) is right-continuous with respect to the Hausdorff topology on compactsets.Brownian motion is a two-dimensional object; if Bt is a Brownian motion in Rd andR(B) = {Bt : t â [0, 1]}, then dim(R(B)) = 2 â§ d. The dimension of the space can play alimiting role in the dimension of R(B) (which occurs in dimension one) but otherwise R(B)is two-dimensional.2The state of super-Brownian motion at a fixed time is two-dimensional. The resultsindicating this take different forms for d = 1, d = 2, and d â¥ 3. First, recall that for d = 1,Xt is absolutely continuous. Its size, in the sense of dimension, is as large as the space willallow.Now consider d â¥ 3. For an increasing, continuous function Ï : R+ â R+, we will writeÏâm(Â·) to denote the Ï-Hausdorff measure on Rd. Let Ï2(r) = r2 log+ log+(râ1). Supposed â¥ 3. Then there exist constants 0 < c(d) < C(d) such that PXX0-a.s., for all Borel sets B2This fact has a beautiful extension called Kaufmanâs Theorem, or the dimension doubling theorem. ForA â [0, 1], let B(A) = {Bt : t â A}, where Bt is a Brownian motion in dimension d â¥ 2. Then for allA â [0, 1], dim(B(A)) = 2dim(A) almost surely. What is most striking is that this holds simultaneouslyfor all sets A, rather than first fixing A, which allows one to apply it to random sets which can themselvesdepend on the Brownian motion.19and all t > 0,c(d)Ï2 âm(B â© S(Xt)) â¤ Xt(B) â¤ C(d)Ï2 âm(B â© S(Xt)). (1.3.11)This very precise result is due to Perkins [79]. It should be emphasized that it holds for allt simultaneously. For fixed t, (1.3.11) holds with c(d) = C(d) almost surely (see Theorem5.2 of [15]).The situation for d = 2 is similar. The lower bound in (1.3.11) holds unchanged. Inthe upper bound, Ï2 âm is replaced with Ïâ2 âm, where Ïâ2(r) = r2(log+ râ1)2. For fixedt > 0, it was shown by Le Gall and Perkins [60] that Xt(Â·) = cÏË2 âm(Â· â© S(Xt)) a.s. for adeterministic constant c > 0, where ÏË2(r) = r2 log+(râ1) log+ log+ log+(râ1).For both d = 2 and d â¥ 3, the above results imply that S(Xt) is Lebesgue null, andso Xt is singular with respect to d-dimensional Lebesgue measure. They also imply thatdim(S(Xt)) = 2. Furthermore, the fact that Xt is equivalent (either up to multiplicativeconstants or exactly) to the Ï-Hausdorff measure on its support for some Ï suggests thatthe study of Xt can fundamentally be reduced to the study of S(Xt). In [81], it is shownthat, with probability one, for all t > 0 Xt can be recovered as the limit of the (suitablyrenormalized) Lebesgue measure on a sequence of shrinking enlargements of S(Xt).Many interesting path and support properties concern the process as a whole ratherthan at a fixed time. This includes the study of polar sets and multiple points (see [14] and[80]). The support of the integrated process satisfies dim(S(Xt)) = 4â§d and Xt is absolutelycontinuous (under mild conditions on X0) for d â¤ 3 [89]. Section III of [83] gives a goodoverview of support properties of super-Brownian motion, including some mentioned above.Let us now consider the (α, β)-superprocess Xt. In some respects, the situation couldnot be more different. This is largely due to the discontinuity and long-range jumps of theα-stable process. The following is originally due to Perkins [80] (see also Evans and Perkins[24]), with more recent work in a more general setting by Li and Zhou [61].Theorem 1.3.4. For X0 âMF (Rd) and t > 0,PXX0(S(Xt) = Rd | Xt 6= 0) = 1.Conditioned on survival, the support of Xt is unbounded and effectively non-random.This property is called instantaneous propagation, because when X0 has compact support,the support of Xt equals Rd a.s. for any t > 0 unless Xt = 0.This theorem also implies that, in contrast with super-Brownian motion, dim(S(Xt)) = da.s. on {Xt 6= 0} regardless of the ambient dimension d. This should not be mistaken asimplying that Xt has simple or uninteresting support properties. What it really implies isthat, unlike super-Brownian motion, the fractal properties of Xt are not well represented by20its closed support. S(Xt) = Rd means that every neighbourhood of every point has positivemass, but it does not tell us how this mass is distributed.A Borel set A â Rd is a Borel support for µ if µ(Ac) = 0. It was shown by Perkins[78] that when d > α and β = 1, Xt has a Borel support Ît with dim(Ît) = α. Moreover,for ÏÎ±(r) = rα log+ log+ râ1, we have the following: there exist universal constants 0 <c(α, d) < C(α, d) such that PXX0-a.s., for all t > 0 there exists a Borel set Ît such that forall Borel B,c(α, d)ÏÎ± âm(B â© Ît) â¤ Xt(B) â¤ C(α, d)ÏÎ± âm(B â© Ît). (1.3.12)So Xt is equivalent up to universal constants to its ÏÎ±-Hausdorff measures restricted to aparticular Borel support Ît (which Perkins defines explicitly) of dimension α. Borel supportsare highly non-unique, but this is the best case scenario, since Theorem 1.3.4 implies thatno such result can hold with the closed support. When d < α, the limiting effect of theambient dimension takes over; in this case, Xt is absolutely continuous with respect to theLebesgue measure.For β < 1 the dimension of Xt is αβ , but the known results are less precise. In [12],Dawson gives a proof that Xt has a Borel support of dimension αβ , and for any randomBorel set A â Rd, Xt(A) = 0 a.s. on {dim(A) < αβ }. See Theorem 7.3.1 of [12]. As we havenoted, Xt is absolutely continuous if and only if d < αβ , as we discuss presently.1.3.5 DensitiesWe now discuss absolute continuity of Xt and Xt, which we have alluded to in the previoussection. Let 0 < α â¤ 2 and 0 < β â¤ 1 and consider the (α, β)-superprocess. (This includesthe case (α, β) = (2, 1), which is super-Brownian motion.) The following is originally dueto Fleischmann [27].Theorem 1.3.5. For t > 0, Xt is absolutely continuous almost surely if and only if β < αd .Thus when β < αd , the superprocess has a density Xt(x) so that Xt(dx) = Xt(x)dx. Forsuper-Brownian motion, this is the case if and only if d = 1. The new results in this thesisconcern properties of the density and so hold in the β < αd regime.Given a density function, it is natural to ask about its regularity properties. We firstdiscuss the regularity of the density of super-Brownian motion in dimension one, which wedenote by Xt(x). In this case, there is a version of the density which is jointly continuousin (t, x). Moreover, it is η-Ho¨lder continuous for all η < 12 in space and all η <14 in time.That is, for each ηx <12 and ηt <14 , for δ > 0 there exists a random constant c(δ) > 0 such21that if |tâ s| < δ and |xâ y| < Î´,|Xt(x)âXs(y)| < c(δ)(|x â y|ηx + |tâ s|ηt). (1.3.13)The modulus of continuity for Xt(x) is often proved as a corollary of another importantresult. It was shown by Konna and Shiga [45], and independently by Reimers [84], thatXt(x) is a jointly continuous solution to the stochastic partial differential equation (SPDE)ââtXt(x) =12âXt(x) +12âXt(x)WË (t, x). (1.3.14)In the above, WË is a space-time white Gaussian noise. (Equivalently, W is a cylindricalWiener process on R+ÃR and WË is its distributional derivative.) The proof that the densityis jointly continuous uses a space-time Kolmogorov continuity criterion which implies theHo¨lder moduli in (1.3.13).The SPDE (1.3.14) is of course formal, as its solution is non-differentiable. For a precisedefinition of a solution and a proof that the equation is solved by the density of super-Brownian motion, see [45] or Section III.4 of [83]. The notes of Walsh [91] are an excellentintroduction to the classical theory of SPDE.We now consider the density when β < 1. Because the stable branching mechanismoccasions birth events when an atom is added to the measure, there is a random set oftimes at which Xt is not absolutely continuous. Like the jumps of a stable process, the setof jump times is countable and dense, so the density cannot be jointly continuous. Mostresults concern the fixed-time regularity of Xt(x).The regularity of super-Brownian motion with (1+β)-stable branching was first studiedby Mytnik and Perkins in [71]. As this is the (α, β)-superprocess with α = 2, the densityexists if and only if d < 2β . They established a dichotomy: if d = 1, the density is continuousat a fixed time; if d > 1 and the density exists, it is locally unbounded. In this work theoptimal modulus of continuity was not considered. It was also shown in [71] that for fixedx â Rd, Xt(x) is locally unbounded in the temporal variable for all 1 â¤ d < 2β .The above results have since been extended to cover all α â (0, 2] and include moduliof continuity. See [76] for a comprehensive survey. The regularity properties are morenuanced for superprocesses with stable branching. We emphasize three articles in particular[28, 29, 75].In [28], the continuity/local unboundedness dichotomy was extended for all α â (0, 2],and the optimal modulus of continuity was determined.Theorem 1.3.6. (a) If d = 1 and α > 1 + β, then there is a version of the density Xt(Â·)which is locally η-Ho¨lder continuous for all η < ηc :=α1+β â 1.22(b) If d > 1, or d = 1 and α â¤ 1 + β, then â1UXt(Â·)ââ = â for all open sets U almostsurely on {Xt 6= 0}.Part (a) above is optimal; it was also shown that Xt(x) is not η-Ho¨lder continuousfor η â¥ ηc. Thus ηc is the optimal local modulus of continuity. The optimal modulus ofcontinuity at a fixed point is better and was established in [29]. We say that f(x) is η-Ho¨ldercontinuous at a point x0 if there exists δ > 0 such thatsupx: |xâx0|â¤Î´|xâ x0|âÎ·|f(x)â f(x0)| <â.Theorem 1.3.7. Let d = 1, α > 1 + β, and fix t > 0 and x â R. Then for any η < Î·âc :=min{1+α1+β â1, 1}, the continuous version of Xt(x) is almost surely η-Ho¨lder continuous at x.The above modulus is again optimal. Finally, the so-called multifractal spectrum ofmoduli of continuity was determined in [75]. For all η â [ηc, Î·âc ), there is a set of points atwhich the density has optimal modulus of continuity η. The Hausdorff dimension of thisset of points was computed for each η.Let us briefly revisit the Laplace functional in the context of absolute continuity. Forsimplicity of notation we will consider Xt only, but the same principle applies for Xt. Theonly requirement is that the density exists. We recall thatEXX0(exp(âXt(Ï))) = exp(âX0(uÏt ))for all Ï â B+b , where uÏt (x) is the unique solution to the evolution equation (1.3.5) with u0 =Ï. (It is also the unique weak solution to (1.3.6) with u0 = Ï.) The dual pairing betweenfinite measures and bounded functions make B+b the appropriate class of test functions tointegrate against Xt when it is a measure. When Xt has a density, we haveXt(Ï) =â«Xt(x)Ï(x) dx.In the above, we can plausibly replace Ï(x)dx with a measure µ(dx) in order to consider thepairing µ(Xt) =â« Xt(x)µ(dx) for certain measures µ. Indeed, this is the case for generalfinite measures, and there exist mild solutions to (1.3.6) with u0 = µ â MF (Rd) [7]. Thedual relationship is preserved, and thus we can analyse functionals of the density like µ(Xt)via solutions to the (1.3.6) with measure initial data.Much of our work on the densities of Xt and Xt involves the analysis of solutions tothe dual equation with measure-valued initial data. This is given the necessary rigoroustreatment in the chapters of the thesis where we prove our results. We defer all discussionsconcerning the existence and uniqueness of solutions with measure-valued initial data and23will refer interchangeably to the solutions of the integral equation and partial differentialequation.1.3.6 The canonical measure and cluster decompositionsEvery infinitely divisible measure has what is called a canonical measure. (For example, seeTheorem 3.4.1 of [12].) It arises, roughly speaking, as the intensity measure in a Poissoniza-tion procedure. In this section we describe the canonical measure and cluster representationsfor superprocesses.First we will consider super-Brownian motion Xt with initial measure X0. By thebranching property, for t > 0 and N â N we can realize Xt as a sum of N independentsuper-Brownian motions with initial measure Nâ1X0. In fact, this holds not just for fixedt but as a process; for all N â N,XÂ·D= XËN,1· + · · Â·+ XËN,NÂ· , (1.3.15)where the XËN,i· are iid with law PXNâ1X0 (see (II.7.1) of [83]). Fix t > 0. A simple argumentusing duality implies thatPXNâ1X0(Xt 6= 0) = pN =: 1â exp(Nâ1X0(1)v(t))where v(t) is the solution to vâ² = â12v2 with v â â as t â 0+. In fact, v(t) = 2/t. Forlarge N , we havepN â¼ Nâ1X0(1)v(t), (1.3.16)where aN â¼ bN means that limNââ aNbN = 1. The number of clusters in (1.3.15) whichsurvive until time t is Binomial(N, pN ). By the Poisson approximation to the binomial and(1.3.16), as N â â the number of surviving clusters converges to a Poisson(X0(1)v(t))random variable.The convergence of the number of surviving clusters in the Nth cluster decomposi-tion (1.3.15) suggests that XÂ· itself has a Poisson representation, and this is indeed thecase. The canonical measure NXx of super-Brownian motion is a Ï-finite measure onD([0,â),MF (Rd)) which, informally, describes the super-Brownian motion issuing froma single infinitesimal ancestor at location x â Rd. It can be obtained via the limitNXx (Â·) = limNââNPXNâ1Î´x(Â·). (1.3.17)(For example, see Theorem II.7.3(a) of [83], which establishes that the renormalized laws ofa sequence of branching particle systems converges to the canonical measure.) The canonical24measure is the intensity measure in a Poisson cluster representation of the superprocess.Let X0 â MF (Rd) and let NXX0 be given byNXX0(Â·) =â«NXx (Â·)X0(dx).We then let Î(Â·) be a Poisson point process on D([0,â),MF (Rd)) with intensity NXX0 .Then the process defined byXt(Â·) =ï£±ï£²ï£³â«Î½t(Â·)Î(dÎ½), t > 0X0(Â·), t = 0(1.3.18)is a realization of super-Brownian motion with initial measure X0, where for a path-valuedâpointâ ν â D([0,â),MF (Rd)) in the point process Î, νt â MF (Rd) is the measure attime t. For a proof, see Theorem II.7.3(a) of [83].The canonical measure is also related to solutions to the dual equation. For Ï â B+b , ifV Ït is the solution to (1.3.2) thenNXx (1â exp(âXt(Ï))) = V Ït (x). (1.3.19)The above can be derived from a general formula for the Laplace functionals of Poisson pointprocesses (for example, see Theorem 24.14 of [43]) and (1.3.1). It can be shown directlyfrom (1.3.19) that the mass of {Xt 6= 0} under NXx is v(t), that is,NXx (Xt 6= 0) =2t. (1.3.20)This is particularly useful when we consider the Poisson cluster representation at a fixedtime. Let Xt be a super-Brownian motion with initial measure X0. From (1.3.18) it followsthatXtD=Nân=1XËit , (1.3.21)where N is a Poisson(X0(1)2tâ1) random variable and the clusters XËit are iid with lawNXX0(Xt â · |Xt 6= 0). For 0 < s < t, conditional on Xs, by the above and the Markovproperty one can express Xt via a cluster decomposition with order Xs(1)(tâ s)â1 clusters.We now consider the (α, β)-superprocess. The heuristics are identical, so we simply statethe relevant results. There is a Ï-finite measure on D([0,â),MF (Rd)) called the canonicalmeasure of the (α, β)-superprocess and denoted NXx such thatNXx (1â exp(âXt(Ï))) = uÏt (x), (1.3.22)25where uÏt (x) is the solution to (1.3.5) with u0 = Ï â B+b . Because the non-linearity of thedual equation is different, the asymptotics of the rate of surviving clusters differ from thesuper-Brownian case. In particular, choosing Ï â¡ λ in (1.3.22) and taking Î»ââ, it followsthatNXx (Xt 6= 0) = Ut, (1.3.23)whereUt =(1βt) 1β. (1.3.24)A Poisson point process realization analogous to (1.3.18) holds in this case as well (forexample see Theorem 4.2.1 of [18]). If Xt has law PXX0 , for fixed t > 0 we haveXt D=Nân=1XË it , (1.3.25)where N is Poisson((Î²t)â1Î² ) and the XË it are iid with law NXX0(Xt â · |Xt 6= 0). The smallerβ is, the more surviving clusters there are in a short time cluster decomposition of Xt. Thisfact is important in the proof of one of our main results about Xt.1.4 Our ResultsWe now describe the new results proved in this thesis. Complete descriptions and proofs aregiven in the research chapters. The discussion is divided into two sections corresponding toour results on super-Brownian motion and the (α, β)-superprocess.In both cases we are interested in the absolutely continuous regime, which we restrictto now. We study fine properties of the density and our analysis is based on the limitingbehaviour of the Laplace functional. Recall the proof that super-Brownian motion has com-pact support which we sketched in Section 1.3.4. Restating the argument using canonicalmeasure, for x â Rd and t, R, λ > 0, we haveNXx (1â (exp(âÎ»Xt(B(0, R)c)) = V λ,Rt (x),where V λ,rt is the solution to (1.3.3) with V0 = λ1B(0,R)c . This implies thatNXx (Xt(B(0, R)c > 0) = Vâ,Rt (x) =: limÎ»ââV λ,Rt (x),so the probability that Xt charges B(0, R)c is governed by the (singular) limit of a familyof solutions to the dual equation.The same principle applies to general charging-type events, i.e. {F (Xt) > 0}, where26F (Xt) = Xt(Ï) for Ï â B+b or F (Xt) = µ(Xt) for µ â MF (Rd), where the quantityµ(Xt) =â«Xt(x)µ(dx) is well-defined when the density exists, as we discuss in Section 1.4.2.Because we are interested in the density, we focus on charging events for measures. Wehave that NXx (µ(Xt) > 0) is equal to limÎ»ââ Vλµt (x), where Vλµt is the solution to the dualequation with initial data λµ. The same holdsmutatis mutandis for the (α, β)-superprocess.While our results for the two processes are very different in nature, the proofs of both havethis principle at their core.In the case of the (α, β)-superprocess we are directly interested in the event{µ(Xt) > 0} for various measures µ, so the connection with the dual equation is obvi-ous. It is a consequence of instantaneous propagation (recall Theorem 1.3.4) that the localproperties of µ determine the probability of {µ(Xt) > 0} to a much greater extent thanits large-scale spatial properties. Our characterization of {µ(Xt) > 0} is based on fractalproperties of µ, in particular on some notions of its dimension, and we prove a dichotomyfor the (α, β)-superprocess concerning when Xt almost surely charges a measure of a givendimension.For super-Brownian motion in dimension one, if µ,X0 â MF (Rd) are both compactlysupported and separated by some distance, the probability that µ(Xt) is positive is largelygoverned by the relative locations of S(X0) and S(µ) because Xt has compact support andpropagates with finite speed. Our interests in this case rather lie with the boundary ofthe zero set of the density, which is closely related to the asymptotics of the probabilityPXX0(0 < Xt(x) â¤ a) for small values of a. PXX0(Xt(x) > 0) is determined by Vât (x),where Vât is the limit as λ â â of solutions to the dual equation with initial data λδ0.Remarkably, the asymptotics of PXX0(0 < Xt(x) â¤ a) are also closely related to Vât , whichwe describe in the next section.1.4.1 A boundary local time for super-Brownian motionWe now discuss our results concerning the density of super-Brownian motion. The work wediscuss here is contained in Chapter 2 and Chapter 3.Let Xt be a super-Brownian motion in dimension one. For t > 0, Xt has a density Xt(x)which is η-Ho¨lder continuous in space for all η < 12 . Furthermore, the density is compactlysupported. Consider the zero set Zt = {x : Xt(x) = 0}. Since Xt has compact support,Zt contains intervals on which Xt(x) is obviously constant. However, these intervals areadjacent to the intervals where the density is positive, which is also where the density isrough (Ho¨lder-12 ). The interface between these sets, which is the boundary of the zero setâZt, is a locus of very delicate behaviour.One of our major results is the construction of a random measure Lt supported onâZt which we call the boundary local time of Xt. The construction of Lt is based on an27approximation scheme via measures whose mass is concentrated where Xt(x) is positive butsmall. Consider the measure Lλt (dx) which is defined by its densityLλt (x) = w(Î»)Xt(x)eâÎ»Xt(x), (1.4.1)where w(λ) is a scaling factor to be determined. The function fλ(z) = zeâÎ»z is mostmassive where z = O(Î»â1), and fλ(0) = 0, and thus the mass of Lλt will concentrate whereXt(x) = O(Î»â1), roughly speaking on {x : 0 < Xt(x) â¤ Î»â1}. Thus, if we take λ â âand choose w(λ) in order to obtain a non-degenerate limit, the limiting measure will besupported on âZt. The limiting measure is Lt.The proper scaling factor w(λ) was known from previous work of Mueller, Mytnik andPerkins [69]. This was the first work dedicated to close analysis of âZt, and it is also the workin which the existence of the boundary local time was conjectured. The main contributionof [69] was to calculate the Hausdorff dimension of âZt. An intermediate result determinedthe left tail of the density when conditioned to be positive. They showed thatPXX0(0 < Xt(x) â¤ a) â a2Î»0â1 (1.4.2)for a â (0, 1]. Here â means equal up to multiplicative constants in a manner that will bemade precise in Chapter 2. There is some dependence on x, t and X0. The value λ0 â (0, 1)is the lead eigenvalue associated to a particular Ornstein-Uhlenbeck operator related to Xt.At the end of this section, after stating our main results, we describe λ0 and how it entersinto the analysis. The left tail (1.4.2) is closely related to the following: for Î»â« 0,EXX0(â«Xt(x)eâÎ»Xt(x)dx)â Î»â2Î»0 .We therefore deduce that in order for Lλt (x) to converge to a finite, non-zero limit as Î»ââ,we should take w(λ) = λ2λ0 . The integral of a test function Ï against of Lλt is thusLλt (Ï) = Î»2Î»0â«Ï(x)Xt(x)eâÎ»Xt(x)dx. (1.4.3)It was shown in [69] that EXX0(Lλt (Ï)) converges as λ â â, and an explicit formula wasgiven for the limit. This led the authors to conjecture that Lλt converges in MF (Rd). Weproved this conjecture in [35].Theorem 1.4.1. PXX0-a.s. there exists a measure Lt supported on âZt such that Lλt â Ltin probability in the metric space MF (Rd), and Lλt (Ï)â Lt(Ï) in L2 for all Ï â Bb(R).We also derive, as a by-product of our method of proof, explicit formulas for first and28second moments of Lt. These are crucial for deriving several other properties of Lt. First,let us revisit dim(âZt). In [69], the authors showed thatdim(âZt) = 2â 2λ0 â (0, 1) (1.4.4)with positive probability on {Xt 6= 0}. More specifically, they showed that dim(âZt) â¤2 â 2λ0 a.s. and dim(âZt) â¥ 2 â 2λ0 with positive probability. The lower bound wasconjectured to hold a.s. on {Xt 6= 0} but this was not shown. Completing the dimensionresult was a major motivation for the construction of the boundary local time.Theorem 1.4.2. Under PXX0 , dim(âZt) = 2â 2λ0 a.s. on {Xt 6= 0}.In order to prove the above using Lt we use Frostmanâs Lemma. For p > 0 the p-energyof µ âMF (Rd) isEp(µ) =â«|xâ y|âpµ(dx)µ(dy). (1.4.5)Frostmanâs Lemma states that a Borel set A supports a non-zero measure µ with finitep-energy if and only if A has positive p-Hausdorff measure. Using the moment formulae wederive for Lt, we show thatEXX0(Ep(Lt)) <â (1.4.6)for all p < 2â 2λ0, which implies that dim(âZt) â¥ 2â 2λ0 whenever Lt 6= 0.To prove Theorem 1.4.2 then requires that Lt(1) > 0 a.s. on {Xt 6= 0}. While a naturaland desirable property for Lt, the proof is non-trivial. It is based on a novel analysis of one-dimensional super-Brownian motion near the right endpoint of its support. The followingwas proved in joint work with Perkins [37].Theorem 1.4.3. Under PXX0 , Lt(1) > 0 a.s. on {Xt 6= 0}.We will sometimes refer to the above as the 0 â 1 law. We also prove several otherresults concerning âZt and Lt. These include the fact that Lt is atomless almost surely anda representation for Lt in terms of the boundary local times associated to each cluster in acluster representation for super-Brownian motion. We also give a local version of the 0â 1law, proving that Lt has positive mass in every neighbourhood of âS(Xt), the boundaryof the support of the density. It follows from an elementary topological argument thatâS(Xt) â âZt. In view of Theorem 1.4.1 and the above, we haveâS(Xt) â S(Lt) â âZt. (1.4.7)Whether these inclusions are strict or are in fact equalities of sets remains unresolved.29We now discuss the measures Lλt and the eigenvalue λ0 in greater detail. Lλt is definedas above because it is amenable to analysis using solutions to the dual equation. We willproceed using canonical measure. First, we observe that for λ > 0,N0(1â exp(âÎ»Xt(x))) = V λt (x), (1.4.8)where V λt is the unique solution toââtVt(x) =12âVt(x)â 12Vt(x)2, V0 = λδ0. (1.4.9)To see that the above holds, one must extend the dual relationship from bounded func-tions to finite measures. As we have indicated, this can be done when the density exists.Differentiating the above with respect to λ yieldsN0(Xt(x) exp(âÎ»Xt(x))) = ââÎ»V λt (x). (1.4.10)Recall that Xt(x)eâÎ»Xt(x) is the non-normalized density of Lλt . Hence the above allows usto analyse moments of Lλt in terms of (the λ-derivative of) Vλt . Formally differentiating(1.4.9) term by term with respect to λ, one sees that Wt(x) =ââÎ»VÎ»t (x) solves the initialvalue problemââtWt(x) =12âWt(x)âWt(x)V λt (x), W0 = δ0. (1.4.11)This differentiation can be made rigorous and in particular ââÎ»VÎ»t (x) is the unique solutionto (1.4.11). This is a linear equation and its solution has a (probabilistic) Feynman-Kacrepresentation. (For a version of the Feynman-Kac formula, see Theorem 7.6 of Karatzasand Shreve [42].) From (1.4.10) and the Feynman-Kac form ula, we haveN0(Xt(x) exp(âÎ»Xt(x))) = ââÎ»V λt (x) = EB0(exp(ââ« t0V Î»tâs(Bs)ds) â£â£â£â£Bt = x), (1.4.12)where Bt is a standard Brownian motion. Via some changes of variables and scaling argu-ments, we can obtain a new expression for the above:N0(Xt(x) exp(âÎ»Xt(x))) = EY0(exp(ââ« log(λ2t)0V es/21 (Ylog(Î»2t)âs)ds) â£â£â£â£Ylog(Î»2t) = x).(1.4.13)Here Y is a standard Ornstein-Uhlenbeck process on R. We will return to the Ornstein-Uhlenbeck process shortly.From (1.4.8), it follows that V λt (x) â¤ N0(Xt 6= 0) = 2/t for all λ > 0. We can therefore30take Î»ââ in (1.4.8) and define Vât (x) byVât (x) = limÎ»ââV λt (x) = NX0 (Xt(x) > 0). (1.4.14)To analyse (1.4.13) we would like to replace Ves/21 with Vâ1 . Clearly we obtain a lowerbound for (1.4.13) if we make this replacement directly. On the other hand, in [69] it wasshown using preliminary estimates on the rate of convergence of V λ1 to Vâ1 thatsupT>0exp(â« T0supyâR|Vâ1 (y)â V es/21 (y)|ds)<â.This implies that, up to multiplicative constants, we haveN0(Xt(x) exp(âÎ»Xt(x))) â EY0(exp(ââ« log(λ2t)0F (Ys) ds)â£â£â£â£Ylog(Î»2t) = x), (1.4.15)where F (y) = Vâ1 (y).To determine the asymptotics of (1.4.15) as Î»ââ we appeal to the spectral theory ofkilled diffusions. Let A be the generator of a standard Ornstein-Uhlenbeck process, Yt, onR. That is, for a twice differentiable test function f ,Af(x) = âx2f â²(x) +12f â²â²(x).For a bounded, continuous, non-negative function Ï, the operatorAÏf(x) = Af(x)â Ï(x)f(x)is the generator of a killed Ornstein-Uhlenbeck process, Y Ït , with killing rate given byÏ. That is, if we continue to denote the unkilled Ornstein-Uhlenbeck process by Yt, theinstantaneous rate of killing at time t is Ï(Yt). If Ï denotes the death time of the process,we can express the survival probability up to time T > 0 conditional on Y asP (Ï â¥ T |Y ) = exp(ââ« T0Ï(Ys) ds). (1.4.16)The operator A is spectrally negative and has discrete spectrum. In particular, there is asequence of eigenvalues 0 â¥ âÎ»Ï0 â¥ âÎ»Ï1 â¥ âÎ»Ï2 â¥ · · · â ââ. It is possible to expressthe transition density (from time 0 to time T > 0) of the killed diffusion in terms of an31eigenfunction expansion which shows thatP (Ï â¥ T ) â eâÎ»Ï0T . (1.4.17)We observe that (1.4.15) can be interpreted as a survival probability as in (1.4.16) withkilling function F . One must handle the conditioning on the endpoint, but it essentiallyfollows from the (1.4.15), (1.4.16) and (1.4.17) thatN0(Xt(x) exp(âÎ»Xt(x))) â (Î»t)â2Î»0 , (1.4.18)where λ0 = λF0 , whence we derive the scaling factor w(λ) = λ2λ0 for Lλt . The asymptoticsof Lλt (x) as Î»ââ are closely related to the left tail of Xt(x). The derivation of (1.4.2) in[69] uses (1.4.18) and a Tauberian theorem.The sketch above indicates how λ0 appears in the analysis of the first moments of Lλtand the left tail of Xt(x) (and therefore of âZt). This first appeared in [69]. Our work, inparticular the proof of Theorem 1.4.1, is based on the convergence of second moments ofLλt , i.e. moments of the form N0(Lλt (Ï)LÎ»â²t (Ï)) for Ï,Ï â B+b and λ, Î»â² > 0. Note thatLλt (Ï)LÎ»â²t (Ï) = (Î»â²Î»)2Î»0â«â«Xt(x)Xt(y)Ï(x)Ï(y) exp(âÎ»Xt(x)â Î»â²Xt(y))dxdy.To compute the expectation of the above, the essential quantity isN0(Xt(x)Xt(y) exp(âÎ»Xt(x)â Î»â²Xt(y))).We can study this in a similar fashion to (1.4.10) using duality and differentiating withrespect to λ and Î»â². We haveN0(Xt(x)Xt(y) exp(âÎ»Xt(x)â Î»â²Xt(y))) = â2âÎ»âÎ»â²VÎ»Î´x+Î»â²Î´yt (0). (1.4.19)Here, VÎ»Î´x+Î»â²Î´yt is the solution to the dual equation with V0 = λδx + Î»â²Î´y. These two-pointed solutions (and their limits as λ, Î»â² ââ) are critical to understanding the limitingbehaviour of second moments of Lλt . The second derivative in (1.4.19) admits a probabilisticrepresentation analogous to but more complicated than (1.4.12), which involves a branchingBrownian path. This can then be rescaled to obtain Ornstein-Uhlenbeck processes as wasdone for first moments. However, establishing the asymptotic size (i.e. (Î»â²Î»)â2Î»0) of (1.4.19)does not suffice for our purposes; to show that Lλt converges in the space of measures inprobability, we need to explicitly compute the limit of (1.4.19) as λ, Î»â² â â, a technicalcomputation of considerable length. In doing so, we obtain a second moment formula for32the boundary local time which we use to show many of our other results, including the finiteenergy result (1.4.6).In parallel with our work [35, 37] (the latter with Perkins) and that of Mueller, Mytnikand Perkins [69] on the boundary of the zero set of Xt, the boundary of the range of super-Brownian motion has been considered. We recall from Section 1.3.4 that the integratedsuper-Brownian motion X(Â·) = â«â0 Xt(Â·)dt has a density X(x) in dimensions d = 1, 2, 3.Mytnik and Perkins computed the Hausdorff dimension of the boundary of the zero setof X(x) in [73]. With Hong, they improved this result by showing that â(S(X)) has thesame dimension [33]. (This remains open at fixed time; that is, it has not been shown thatâ(S(Xt)) has the same dimension as â(Zt).) Very recently and in the spirit of our work,Hong [34] has constructed a boundary local time supported on â(S(X)); the tools he usesin this higher dimensional setting are quite distinct from those described above.1.4.2 New path properties for the (α, β)-superprocessIn this section our object of study is the (α, β)-superprocess, Xt, on Rd with α < 2 in theabsolutely continuous regime β < αd . We recall from our earlier discussion that for t > 0there is a continuous version of the density Xt(x) if and only if d = 1 and α > 1 + β, andthe density is locally unbounded in the discontinuous case. Remarkably, with one exceptionour results are completely agnostic to this dichotomy.We recall that for all β â (0, 1], the (α, β)-superprocess with α < 2 has instantaneouspropagation: for t > 0, supp(Xt) = Rd almost surely when conditioned on survival. Thusthe event that Xt has positive mass on a set of positive d-dimensional Lebesgue measure istrivial.Our results are in the spirit of instantaneous propagation, in that we classify certaintrivial events for the density. For µ â MF (Rd), we defineµ(Xt) =â«Xt(x)µ(dx). (1.4.20)A priori this object is not well-defined, because the density is only unique up to Lebesguenull sets, and, in view of instantaneous propagation, we are interested in µ(Xt) for measuresµ which are singular with respect to the Lebesgue measure. We show that µ(Xt) is well-defined for any finite measure µ by first arguing that the density exists almost surely at anyfixed point.A natural starting point is to consider µ = δx, so that µ(Xt) = Xt(x). We defineÎ²â(Î±) = αd+α .Theorem 1.4.4. The following hold under NX0 and PXX0 for X0 âMF (Rd).(a) Let β â¤ Î²â(Î±). Then for fixed x â Rd and t > 0, Xt(x) > 0 almost surely on {Xt 6= 0}.33(b) Let Î²â(Î±) < β < αd . For fixed x â Rd and t > 0, Xt(x) = 0 with positive probability on{Xt 6= 0}. Moreover,C1tâ 1β1 + |tâ 1Î±x|d+Î±â¤ NX0 (Xt(x) > 0) â¤ C2tâ 1β log(e+ |tâ 1αx|)1 + |tâ 1αx|d+α(1.4.21)for constants 0 < C1 < C2.The above result is due to Chen, Ve´ron, and Wang [7] for β < Î²â(Î±) and the firsttwo authors for β = Î²â(Î±) [8]. In fact, these authorsâ work was on solutions to the dualequation. Their main theorem, which establishes a dichotomy concerning the existence ofvery singular solutions to the dual equation, is equivalent to the above statement about thedensity of the (α, β)-superprocess.The above theorem and instantaneous propagation characterize the event {µ(Xt) > 0}when µ = δx and µ = mU , the Lebesgue measure restricted to an open set U . Whenconditioned on survival, mU(Xt) > 0 a.s. for all parameters and δx(Xt) = Xt(x) > 0 a.s.if and only if β â¤ Î²â(Î±). The point mass δx and Lebesgue measure on an open set are,locally speaking, the most concentrated and most spread out measures on Rd. Our resultsinterpolate between these two and hold for general measures.For δ â [0, d], let Î²â(Î±, δ) = Î±dâÎ´+Î± . As we will see, this is the critical parameter for ameasure which is âof dimension Î´.â We write Hδ(A) to denote the xδ-Hausdorff measure ofa set A. (See [25] or [62] for an introduction to Hausdorff measure.) The Hausdorff measureand Hausdorff dimension of a measureâs support are closely related to mass conditions onsmall balls for the measure. For example, if µ â MF (Rd) satisfiessupxâRd,r>0râÎ´µ(B(x, r)) <â, (1.4.22)then µ is at least δ-dimensional in the sense that Hδ(S(µ)) > 0, and hence dim(S(µ)) â¥ δ.(This is called Frostmanâs Lemma. It is discussed in the introduction of Chapter 4; see alsoTheorem 8.8 of [62].) A similar result holds for lower bounds: ifinfxâS(µ),0<r<1râÎ´µ(B(x, r)) > 0, (1.4.23)then dim(S(µ)) â¤ δ. The following is a partial statement of our results on this problem.For x â Rd and A closed, we let d(x,A) = infyâA |y â x|.Theorem 1.4.5. Let δ â [0, d] and X0 â Rd. For t > 0 we consider Xt when conditionedon survival under PXX0 and NX0 .(a) Suppose that µ â MF (Rd) satisfies Hδ(S(µ)) > 0. Then if β â¤ Î²â(Î±, δ), µ(Xt) > 034almost surely.(b) Suppose that µ has compact support and satisfies (1.4.23). Then if β > Î²â(Î±, δ), µ(Xt) =0 with positive (conditional) probability. Moreover, for all t > 0 and x â Rd we haveC1tâ 1β1 + |tâ 1α d(x, S(µ))|d+Î±â¤ NXx (µ(Xt) > 0) â¤ C2[tâ1Î²â δα â¨ tâ 1β] log(e+ tâ 1α d(x, S(µ)))1 + |tâ 1αd(x, S(µ))|d+α.(1.4.24)for constants 0 < C1 < C2 which depend only on α, d and µ.The (α, β)-superprocess therefore exhibits remarkably reliable behaviour on a micro-scopic level. The parameters determine a scale beyond which all charging events involvingthe density at a fixed time are trivial. In the non-trivial regime (corresponding to part (b) inthe above theorem), (1.4.24) gives asymptotic bounds on NXx (µ(Xt) > 0) as d(x, S(µ))ââ(for fixed t > 0) and t â 0 (for fixed x). Under additional assumptions on µ, we improve thelower bound in (1.4.24) so that the power of t matches the power of t in the upper boundfor t â¤ 1 (see Theorem 4.1.20 in Chapter 4). In this case our estimates are nearly sharp.The methods we use to prove parts (a) and (b) in Theorem 1.4.5 are completely different.The proof of (a) is probabilistic in nature and has an underlying cluster heuristic. As such,it also gives a new simple probabilistic proof of Theorem 1.4.4(a) by taking µ = δx. To seehow the point case is recovered from our general result, we observe that the dimension δ ofµ = δx is 0 (in particular, H0(S(δx)) = 1 > 0 for part (a), and µ = δx satisfies (1.4.23) withδ = 0 for part (b)) and that Î²â(Î±, 0) = Î²â(Î±).Before discussing the proof of Theorem 1.4.5(a), we expand on the connection betweencharging type events for Xt and the limiting behaviour of the Laplace functional, which wehave alluded to previously. Let λ > 0 and µ â MF (Rd). ThenNXx (1â exp(âµ(Xt))) = uλµt (x), (1.4.25)where uλµt (x) is the unique solution to (1.3.6) with measure-valued initial data λµ. (Apriori, uλµt (x) is a solution to the integral equation when Ï is replaced with λµ; it does, infact, coincide with a weak solution to the (1.3.6). We handle this technicality in the mainbody of the thesis.) We have uλt (x) â¤ Ut for all λ > 0, where we recall from (1.3.23) and(1.3.24) thatNX0 (Xt 6= 0) = Ut =(1βt) 1β. (1.4.26)It follows from (1.4.25) that uâµt (x) = limÎ»ââ uλµt (x) exists and satisfiesuâµt (x) = NXx (µ(Xt) > 0). (1.4.27)35This is in the same spirit as (1.4.14) in the previous section. From (1.4.26) and (1.4.27), weconclude the following:uâµt (x) â¡ Ut ââ µ(Xt) > 0 a.s. on {Xt 6= 0}. (1.4.28)If uâµt (x) â¡ Ut, we say that uâµt is flat. Otherwise we say it is non-flat. The event that Xtcharges µ is trivial (i.e. µ(Xt) > 0 a.s. when Xt is conditioned on survival) if and only ifuâµt is flat.To illustrate the above connection, let us return to Theorem 1.4.4, which concerns thedensity at a fixed point. In [7] the PDE version of Theorem 1.4.4 was motivated by thestudy of very singular solutions (VSS) to (1.3.6). Informally, the VSS is the solution withinitial data equal to â · δ0.3 A standard approach, and the one they use, is to try toconstruct the VSS as the limit of uλt (x), where uλt (x) is the solution ofï£±ï£²ï£³ââtut(x) = âÎ±ut(x)â u1+βt (x)u0 = λδ0.(1.4.29)Let uât (x) = limÎ»ââ uλt (x). In [7] it was found that uât (x) was a very singular solution ifand only if β â (Î²â(Î±), αd ), where we recall that Î²â(Î±) = αd+α . For β â¤ Î²â(Î±), uât (x) â¡ Ut.The claims of Theorem 1.4.4 then follow under canonical measure from (1.4.27) with µ = δ0,since for x â Rd we haveNX0 (Xt(x) > 0) = uât (x). (1.4.30)The results can be extended to Xt under PXX0 via a cluster argument. The dichotomyconcerning the existence of a VSS for the equation (1.3.6) and its probabilistic consequenceswere what originally led to our work on the (α, β)-superprocess. Our work generalizes fromµ = δx to general measures µ and gives a dichotomy in terms of the âdimensionâ of measureas we have noted.We now sketch the proof of Theorem 1.4.5(a) using a cluster argument. Let µ âMF (Rd)and suppose that B is a closed ball of radius 1 such that µ(B) > 0. By instantaneouspropagation, for t > 0, when Xt is conditioned on survival we have Xt(B) > 0 a.s. Byconditioning, one can restrict to the event {Xt(B) â¥ Ç«} for Ç« > 0.We consider P (µ(Xt) > 0 | Fs), where Fs = Ï({Xu : u â¤ s}). Conditional on Fs, by theMarkov property we can realize Xt via a cluster decomposition as in (1.3.25). On the event3The solution Vât (x) from the previous section is the VSS of âtV =12âV â 12V 2, which was originallyshown to exist by Brezis, Peletier and Terman [6]. Their results built on seminal work of Brezis and Friedmanestablishing the existence of solutions with measure-valued initial data [5]. The first work on measure-valuedinitial data and the VSS for the fractional equation was done thirty years after the same for the non-fractionalequation.36{Xt(B) â¥ Ç«}, one can appeal to fixed time continuity results to show that {Xs(B) â¥ Ç«/2}for s sufficiently close to t. Therefore, the intensity of clusters originating in B in the times cluster decomposition of Xt is at least of order Ç«(tâs)â1Î² (recall from (1.3.23) and (1.3.24)that NXx (Xtâs 6= 0) â (tâ s)â1Î² as sâ tâ).Each cluster X i originating in B at time s has some probability that µ(X i) > 0. Wenow suppose that for some δ â [0, d],µ(B(x, r)) â¤ Crδ (1.4.31)for some C > 0 for all x â Rd and r > 0. The condition quantifies how spread out the massof µ is; the larger the value of δ, the larger the set over which µ spreads its mass (note that(1.4.31) corresponds to (1.4.22), a notion that µ is at least δ-dimensional) and hence themore likely it is that a cluster X i charges µ.From (1.4.27), we observe that a lower bound on the probability that a canonical clustercharges µ is equivalent to a lower bound on uâµt (x). (Since we ultimately show that uâµt =Ut, this is a preliminary lower bound.) As in the previous section, we analyse uâµt (x) bydifferentiating uλµt (x) with respect to λ and using the Feynman-Kac formula. In this case,for λ > 0 we obtainââÎ»uÎ»µt (x) =â«EWx(exp(â(1 + Î²)â« t0uÎ»µtâu(Wu)Î² du) â£â£â£â£Wt = y)µ(dy),where Wt is a symmetric α-stable process. From the above, one can see that an upperbound for uλµt (x) can be used in the Feynman-Kac equation to obtain a lower bound forââÎ»uÎ»µt (x), and hence for uâµt (x). In order to bound uλµt (x) above, we use a comparisonwith the solution to the linear fractional heat equation, which itself bounded by analysingcertain potential integrals involving the fractional heat kernel (i.e. the transition density ofthe stable process with index α) and µ. This is of course where the condition (1.4.31) isessential. Ultimately, using this method we show that for all x â B,NXx (µ(Xtâs) > 0 | Xtâs 6= 0) â¥ c(tâ s)âÎ±dâÎ´+Î± (1.4.32)for a constant c > 0. Hence in the time s cluster decomposition of Xt we have order (tâs)â1Î²independent trials (clusters), each with success probability bounded below as in (1.4.32).It follows that the probability of success converges to 1 as sâ tâ when β < Î²â(Î±, δ). Thecase of equality, that is β = Î²â(Î±, δ), is more delicate, but we show that the probabilityconverges to 1 in this case as well.The above argument proves Theorem 1.4.5(a) in the case that µ satisfies (1.4.31). Weare ultimately able to extend this to all measures µ which satisfy Hδ(S(µ)) > 0 by the37following argument. Frostmanâs Lemma, which we discussed above, asserts that a Borelset A â Rd satisfies Hδ(A) > 0 if and only if A supports a measure satisfying (1.4.31).Thus, given a measure µ such that Hδ(S(µ)) > 0, there is a measure µâ² which satisfiesS(µâ²) â S(µ) and (1.4.31), and the proof above applies to µâ². A PDE argument then allowsus to transfer the conclusion from µâ² to µ. This is somewhat surprising, as it implies that itis strictly the closed support of µ, and not how µ locally distributes mass, which determinesif it is charged a.s. by Xt.The proof of Theorem 1.4.5(b) uses a very different argument and is analytical ratherthan probabilistic. It is based on, and is a generalization of, the proof used in [7] to provethat uât is non-flat when β > Î²â(Î±, d) (that is to prove the existence of a VSS for the dualequation). We do not describe it in detail here. Both sides of the dichotomy we prove (i.e.parts (a) and (b) of Theorem 1.4.5) correspond to new existence and non-existence resultsfor solutions to (1.3.6) with very singular initial conditions. Our analytic results belong tothe so-called initial trace theory of (1.3.6), which was introduced in [8] and is still in itsearly stages. These results, as well as several probablistic corollaries of Theorem 1.4.5, arestated in the introduction to Chapter 4.We conclude the section with the statement of another new theorem. By Theorem 1.4.4,if β â¤ Î²â(Î±) then Xt(x) > 0 for a.e. x â Rd when Xt 6= 0. It is natural to ask if this can beimproved from a.e. x to all x â Rd. We show that when the density is continuous, this isthe case.Theorem 1.4.6. Suppose that d = 1, α > 1 + β and β < Î²â(Î±). Then with probability oneon {Xt 6= 0},Xt(x) > 0 for all x â R.The proof combines the tools used in our (probabilistic) proofs of of Theorem 1.4.4(a)and Theorem 1.4.5(a) with the regularity of the density established by Fleischmann et. al[28].1.5 Notes on the textThe material on super-Brownian motion is divided into two chapters, Chapter 2 and Chap-ter 3, which correspond to the articles [35] and [37], respectively. Roughly speaking, weconstruct the boundary local time of super-Brownian motion (Theorem 1.4.1) in Chapter 2and prove that it is non-zero almost surely when conditioned on survival (Theorem 1.4.3)in Chapter 3. Because both of these are required to prove Theorem 1.4.2, this result canbe seen as being proved over both chapters. It is stated in Chapter 3.The material on the (α, β)-superprocess is contained in Chapter 4.38We have done our best to ensure that, where possible, the notation in these chaptersmatches the notation used in the introduction. For any discrepancies, the notation definedwithin the chapter should be used.39Chapter 2The boundary local time ofsuper-Brownian motion I:construction and propertiesSummary. For a one-dimensional super-Brownian motion with density X(t, x), we con-struct a random measure Lt called the boundary local time which is supported on BZt :=â{x : X(t, x) = 0}, thus confirming a conjecture of Mueller, Mytnik and Perkins [69]. Ltis analogous to the local time at 0 of solutions to an SDE. We establish first and secondmoment formulas for Lt, some basic properties, and a representation in terms of a clusterdecomposition. Via the moment measures and the energy method we give a more directproof that dim(BZt) = 2 â 2λ0 > 0 with positive probability, a recent result of Mueller,Mytnik and Perkins [69], where âÎ»0 is the lead eigenvalue of a killed Ornstein-Uhlenbeckoperator that characterizes the left tail of X(t, x). In Chapter 3, we use the boundary localtime and some of its properties proved here to show that dim(BZt) = 2 â 2λ0 a.s. on{Xt(R) > 0}.402.1 Introduction & statement of main resultsSuper-Brownian motion is a Markov process taking values in the space of finite measureson Rd, MF (Rd), equipped with the topology of weak convergence. We denote this processby X = (Xt : t â¥ 0) and denote by PXX0 and EXX0 , respectively, a probability and itsexpectation under which X is a super-Brownian motion with initial data X0 â MF (Rd). Inone dimension, Xt is almost surely an absolutely continuous random measure and thus hasa density we denote by X(t, x). The density is jointly continuous (and will exist) for t > 0,and is continuous with Ho¨lder index 12 â Ç« in the spatial variable for all Ç« > 0 (see [83], forexample, where this is implicit in the proof of Theorem III.4.2). It was shown by Konnoand Shiga in [45] and independently by Reimers in [84] that X(t, x) satisfies the followingstochastic partial differential equation (SPDE):âX(t, x)ât=âX(t, x)2+âX(t, x)WË (t, x), (2.1.1)where WË (t, x) is a space-time white noise. For a complete discussion of such equations,including the precise definition of a solution, see [91] and [45].Before discussing our results, we give a brief introduction to the canonical measure N0of super-Brownian motion. N0 is a Ï-finite measure on C([0,â),MF (R)){0}. It describesthe behaviour of a single cluster, that is, the descendants of a single ancestor, of super-Brownian motion started at the origin. (Likewise Nx is a cluster started from x and isjust a translation of N0.) In fact, one way of obtaining N0 is as a weak limit of branchingparticle systems starting with a single particle, as in Theorem II.7.3 of [83]. Although N0itself is an infinite measure, when restricted to {Xt > 0} for t > 0 it is finite; in particularwe have N0({Xt > 0}) = 2/t (see Theorem II.7.2 of [83]). A fact of central importanceabout the canonical measure is that super-Brownian motion under PXX0 can be understoodas a superposition of canonical clusters. This is discussed in greater detail later on (see(2.1.15)). We will use the notation Xt and X(t, x) to denote the superprocess and itsdensity, respectively, under both PXX0 and N0. The law of the process will always be clearfrom context. For a complete overview of the canonical measure, including proofs of theproperties just stated, see Section II.7 of [83].In a recent work by Mueller, Mytnik and Perkins [69], the authors studied the small-scale asymptotic behaviour of X(t, x), as well as the boundary of its zero set. We define therandom set Zt = {x â R : X(t, x) = 0}. The boundary of the zero set BZt is then definedasBZt := âZt = {x â Zt : (xâ Ç«, x+ Ç«) â© Zct 6= â âÇ« > 0},where the second equality holds by continuity of the density. The results in [69] involve an41eigenvalue λ0 â (12 , 1) which we describe in greater detail shortly. The authors of [69] showthat the left tail of the distribution of X(t, x) behaves likePXX0(0 < X(t, x) < a) â tâ1/2âÎ»0 a2Î»0â1 (2.1.2)as a â 0, where f(a) â g(a) means that f(a) is bounded above and below by cg(a) fordifferent constants c. Clearly for the above to be true we must take t â¥ t0 for some t0 > 0.The upper bound is uniform in x and the lower bound required a localizing assumption.For details, see Section 4 and in particular Theorem 4.8 of [69]. Let dim(B) denote theHausdorff dimension of a set B â R.Theorem 2.A. [Mueller, Mytnik, Perkins (2017)] Under PXX0 , dim(BZt) â¤ 2â 2λ0 almostsurely on {Xt > 0} and dim(BZt) â¥ 2â 2λ0 with positive probability.Because λ0 â (1/2, 1), the dimension satisfies 2 â 2λ0 â (0, 1). The lower bound wasconjectured to hold with full probability on {Xt > 0}, implying that dim(BZt) = 2 â 2λ0almost surely on {Xt > 0}. The difficulty in proving that the lower bound for the dimensionholds with probability one on {Xt > 0} is owing to the delicate nature of the BZt. It is notmonotone in the initial conditions nor in the measure Xt itself.We will construct a random measure Lt, which we call the boundary local time of Xt,supported on BZt. (See Theorems 2.1.1 and 2.1.2.) The existence of Lt was conjectured inSection 5.1 of [69]. Once we have constructed Lt, we use it to give a simpler alternative proofof the lower bound in Theorem 2.A. Our method is to show that Lt has finite p-energy forall p < 2â 2λ0; in particular, see Theorem 2.1.3 below. In Chapter 3, Lt and several of itsproperties derived here, including Theorem 2.1.2(a), Proposition 2.1.6 and Theorem 2.1.9,will be used to resolve the problem left open in Theorem 2.A and Theorem 2.1.3, showingthat dim(BZt) = 2â 2λ0 almost surely on {Xt > 0}.We now give a description of λ0. Define a function F (x) byF (x) := â logPXδ0({X(1, x) = 0}) = N0({X(1, x) > 0}) > 0. (2.1.3)The second equality is standard and is a consequence of (2.1.14) below. Section 2.3, from(2.3.5) to (2.3.14), provides a thorough overview of F as the limit as Î»ââ of the familyof functions {V λ1 }λ>0 which characterize the Laplace transform of the density X(t, x). LetAf(x) = 12fâ²â²(x)â x2f â²(x) denote the infinitesimal generator of a standard, one-dimensionalOrnstein-Uhlenbeck process Y . For a bounded, continuous and non-negative function Ï withlimits at infinity (F is such a function), AÏf = Af â Ïf is the generator of an Ornstein-Uhlenbeck process with Markovian killing corresponding to Ï; that is, for a sample path{Ys : s â [0,â)} â C([0,â);R), we define the lifetime of the process as ÏÏ, after which it42is âkilled,â or put into an inert cemetery state. The distribution of ÏÏ is given byP (Ï > t |Y ) = exp(ââ« t0Ï(Ys) ds)for t > 0. (2.1.4)Section 2.2 develops the relevant theory for these processes and their generators. In partic-ular, Theorem 2.2.1 states that AÏ, taken as an operator on the appropriate Hilbert space,has a countable orthonormal family of eigenfunctions {ÏÏn}ân=0 with corresponding discretespectrum 0 â¥ âÎ»Ï0 â¥ âÎ»Ï1 â¥ · · · â ââ. We define λ0 = λF0 > 0. As we have noted, itwas shown in [69] that λ0 â (1/2, 1). Numerical estimates by Zhu [96], for which the stateddigits are expected to be accurate, suggest that λ0 â 0.8882. This implies that the value ofdim(BZt) from Theorem 2.A, 2â 2λ0, is approximately 0.224. A more detailed discussionof the numerics can be found in the introduction of Chapter 3.The method the authors of [69] used to show (2.1.2) involved computing the asymptoticbehaviour of the Laplace transform of the density. In particular (see Proposition 4.5 of thatwork),limÎ»ââtÎ»0 λ2λ0 EXX0(â«Ï(x)X(t, x) eâÎ»X(t,x)dx)= c0â«â«Ï(w0 +âtz) exp(â1tâ«F (z + tâ1/2(w0 â x0) dX0(w0))ÏF0 (z) dm(z) dX0(x0)(2.1.5)for every bounded Borel function Ï, where m(dz) denotes the unit variance Gaussianmeasure in one dimension, c0 is a positive constant and ÏF0 is the lead eigenfunction ofAF . For a super-Brownian motion with density X(t, x), for λ > 0 we define the measureLλt â MF (R) by dLλt (x) = Î»2Î»0eâÎ»X(t,x)X(t, x) dx. That is, for a bounded measurablefunction Ï : Râ R, we defineLλt (Ï) = Î»2Î»0â«Ï(x)X(t, x) eâÎ»X(t,x)dx. (2.1.6)Lλt is defined the same way under PXX0and N0. The scaling factor of λ2λ0 can be deducedfrom (2.1.5). The convergence of EXX0(Lλt (Ï)) as λ â â, noted in (2.1.5), led the authorsof [69] to conjecture (Section 5.1 of that reference) that there is a random measure Lt onR such that Lλt â Lt in MF (R) in probability. Our main result is the verification of thisconjecture. In all that follows, X0 â MF (R).Theorem 2.1.1 (Boundary local time: existence and convergence). Let t > 0. Under bothPXX0 and N0 there is a random measure Lt(dx) â MF (R), supported on BZt, such thatLλt â Lt in measure as λ â â, and there is a sequence λn â â such that Lλnt â Lt43a.s. as n â â. Moreover, under PXX0 or N0, for all bounded and continuous functions Ï,LÎ»t (Ï)â Lt(Ï) in L2 as Î»ââ.By almost sure convergence, we mean that for PXX0 or N0-a.e. Ï, Lλnt (Ï)â Lt(Ï) weaklyin MF (R). Convergence in measure means with respect to any metric on MF (R) whichinduces the weak topology, e.g. the Wasserstein metric (see for example p. 48 of [82]).Theorem 2.1.2 (Properties of Lt). (a) For all t > 0, we have PXX0(Lt > 0 |Xt > 0) > 0and N0(Lt > 0 |Xt > 0) â¥ 1âÎ»02 .(b) Lt is atomless almost surely under PXX0and N0.Definition. Lt is the boundary local time of Xt.We note that Zt will contain intervals, unlike the zero set of a Brownian motion (whichis equal to its boundary). It is easy to see that Lt is supported on BZt from the factthat as λ gets large, Lλt concentrates on {x : 0 < X(t, x) = O(Î»â1)}, and properties ofthe weak topology on MF (R) (see the proof of Theorem 2.1.1 in Section 2.4). For fixedt > 0, x â X(t, x) is a continuous path taking values in R+ = [0,â). BZt is the set ofpoints where this path begins and ends its excursions from 0. As Lt is supported on BZt,in this sense Lt is a local time of xâ X(t, x) on these excursion endpoints, and hence theboundary local time of X(t, Â·).The existence of a measure supported on BZt allows us to use the energy method tostudy its dimension. We will provide a second moment formula for Lt, with which wecompute the expectation of energy integrals of the formâ«â«|xâ y|âp dLt(x) dLt(y). (2.1.7)If Lt > 0 and the above energy is finite, then dim(supp(Lt)) â¥ p by Frostmanâs connectionbetween energy integrals and Hausdorff dimension (see Theorem 4.27 of Mo¨rters and Peres[65]). We introduce some notation. For h : R2 â R, define (Lt Ã Lt)(h) by(Lt Ã Lt)(h) =â«â«h(x, y) dLt(x) dLt(y).For p > 0, we define hp(x, y) = |xâ y|âp. The second moment formula for Lt allows us toestablish the following.Theorem 2.1.3 (Finite energy and Hausdorff dimension). Both EXX0((Lt Ã Lt)(hp)) andN0((Lt Ã Lt)(hp)) are finite for all p < 2 â 2λ0. Moreover, dim(BZt) = 2 â 2λ0 almostsurely on {Lt > 0} under both measures.44The fact that dim(BZt) â¤ 2 â 2λ0 PXX0-a.s. is already known from Theorem 2.A, andfrom this it follows easily under N0, as we point out in the proof of Theorem 2.1.3. By theabove, the lower bound, i.e. dim(BZt) â¥ 2 â 2λ0, holds with at least the probability thatLt > 0, as in Theorem 2.1.2(a). This plays an important role in Chapter 3; in Theorem3.1.2 of that chapter we show that with respect to both PXX0 and N0, Lt > 0 almost surelyon {Xt > 0}, thus improving part (a) of Theorem 2.1.2 above and establishing almost surenon-degeneracy of Lt. Combined with Theorem 2.1.3, this completes the dimension result.There are a number of other potential uses for such a local time. We now discusssome possibilities. By sampling a point from Lt, we are able to âview Xt from the per-spective of a typical point in BZt.â More precisely, one can define QX0((Z,Xt) â A) =EXX0(â«1A(z,Xt) dLt(z)) and study properties of the Palm measure QX0(Xt â · |Z = z).The behaviour of Xt near BZt is complex and there is still much that is not understoodabout it. For example, the density has an improved modulus of continuity and is nearlyLipzschitz (i.e. Ho¨lder 1â η for all η > 0) at points in BZt (see Theorem 2.3 of [72]). Thissuggests that BZt would be small, but despite this BZt has positive dimension. Construct-ing and studying the Palm measure described above would give a more structured approachfor investigating this phenomenon.As a local time, Lt has the potential to study pathwise uniqueness in the SPDE (2.1.1),a problem which remains open, assuming a similar role as that of the semi-martingale localtime in the Yamada-Watanabe Theorem for one-dimensional SDEs (see Theorem V.40 ofRogers and Williams [86]). It may also provide insight in the behaviour of some discreteprocesses; super-Brownian motion in high dimensions is the scaling limit of a number oflattice models and interacting particle systems. In dimension one, it is still the scaling limitof branching random walk (for example see [92] or Theorem II.5.1(iii) of [83]). One couldobtain information about the boundaries of such approximating processes by proving a limittheorem establishing weak convergence of the laws of their discrete local times to that of Lt.Of course, Lt allows for us to study BZt more directly, as we have done in Theorem 2.1.3.In fact, with Lt it may be possible to determine the exact Hausdorff measure function ofBZt.We now discuss the method of our proof. Upper bounds on second moments of Lλtwere obtained in Section 5.1 of [69], but in order to establish the existence of Lt we requireexact asymptotics, which are more delicate. The main ingredient is the following conver-gence result. In order to state it we need to introduce some notation. Recall that m(dx)denotes the centred unit variance Gaussian measure. Let Ï0 = ÏF0 (the eigenfunction ofAF corresponding to eigenvalue âÎ»0). The constant C2.1.4 is given explicitly in (2.5.46),and the function Ï is defined in (2.5.47). The function Vâ,ât is defined in Section 2.3 asVâ,ât (x1, x2) = N0 ({X(t, x1) > 0} âª {X(t, x2) > 0}) (see (2.3.25)). Finally, we will denote45by PBx and EBx the law and associated expectation of a standard Brownian motion Bt withinitial value B0 = x.Theorem 2.1.4 (Convergence of second moments of Lλt ). There exists a constant C2.1.4 >0 and continuous function Ï : RÃ Râ (0, 1] such that for bounded Borel h : R2 â R,limÎ»,Î»â²ââN0((LÎ»t Ã LÎ»â²t )(h))= C22.1.4â« t0(tâ s)â2Î»0[ â«â«EB0(h(âtâ s z1 +Bs,âtâ s z2 +Bs))Ã exp(ââ« s0Vâ,âtâu (âtâ s z1 +Bs âBu,âtâ s z2 +Bs âBu) du)Ã Ï(z1, z2)Ï0(z1)Ï0(z2) dm(z1) dm(z2)]ds.Moreover, the limit is finite for all bounded h.That the formula above is finite is not obvious, as λ0 > 1/2; we discuss this in moredetail shortly. From the above we can deduce that {Lλt (Ï)}Î»>0 is Cauchy in L2(N0) andtherefore has a limit by completeness; in particular see Corollary 2.4.1 and its proof. Wethen argue that the limit is in fact the integral with respect to a unique measure, whichis Lt. The proof of Theorem 2.1.4 is long and technical; Section 2.5 is entirely devotedto it. We use the Laplace functional to obtain a Feynman-Kac type representation forN0(Lλt (Ï)LÎ»â²t (Ï)) and then establish its convergence. The reason we do so under N0 isbecause the Feynman-Kac formulas are simpler in this setting. We now present first andsecond moment formulas for Lt under N0; as one would expect, the second moment formulain part (b) agrees with the limit of N0((Lλt Ã LÎ»â²t )(h)) given in Theorem 2.1.4. The termsC2.1.4 and Ï are the same that appeared in that result.Theorem 2.1.5 (Moments of Lt under N0). (a) For a bounded or non-negative Borelfunction Ï : Râ R,N0(Lt(Ï)) = C2.1.4 tâÎ»0â«Ï(âtz)Ï0(z) dm(z). (2.1.8)46(b) For measurable h : R2 â R, either bounded or non-negative,N0((Lt Ã Lt)(h))= C22.1.4â« t0(tâ s)â2Î»0[ â«â«EB0(h(âtâ s z1 +Bs,âtâ s z2 +Bs))Ã exp(ââ« s0Vâ,âtâu (âtâ s z1 +Bs âBu,âtâ s z2 +Bs âBu) du)Ã Ï(z1, z2)Ï0(z1)Ï0(z2) dm(z1) dm(z2)]ds. (2.1.9)Moreover, (2.1.9) is finite for all bounded h.As we noted earlier, finiteness of (2.1.9) is not obvious since λ0 > 1/2 (although it isimplicit in the proof of Theorem 2.1.4), which can make (2.1.9) hard to use; for applications,the following upper bound for second moments is easier to apply than the exact formula.The value θ is defined as θ =â«Ï0 dm. Y is an Ornstein-Uhlenbeck process started atz1 with corresponding expectation EYz1 . The exponential term in the first bound of thefollowing proposition can be interpreted as a survival probability of Y , producing a wλ0term which makes the integral finite. (The proofs of Theorem 2.1.3 and Theorem 2.1.2(b)in Section 2.4 both use this technique.)Proposition 2.1.6 (Second moment bounds under N0). For a non-negative Borel functionh : R2 â R,N0((Lt Ã Lt)(h)) â¤ C22.1.4â« t0wâ2Î»0[ â«â«EYz1(exp(ââ« log(t/w)0F (Yu) du)Ã h(âtYlog(t/w),âtYlog(t/w) +âw(z2 â z1)))Ï0(z1)Ï0(z2) dm(z1) dm(z2)]dw.(2.1.10)Moreover,N0(Lt(1)2) â¤ C22.1.4Î¸21â λ0 t1â2Î»0 . (2.1.11)As we have alluded to, applying (2.1.10) with h(x, y) = |xây|âp gives an upper bound forthe expectation of energy integrals of the form (2.1.7), which is how we prove Theorem 2.1.3.Thus far, we have not commented on the proofs of existence and properties of Lt underPXX0 . The proofs rely on the conditional representation in terms of canonical clusters, whichwe will discuss shortly. First, in order to keep the moment results together, we state ourresults regarding the moments of Lt under PXX0.47Theorem 2.1.7 (Moments of Lt under PXX0). For a bounded or non-negative Borel functionÏ : Râ R,EXX0(Lt(Ï)) = C2.1.4 tâÎ»0â«â«Ï(x0 +âtz) exp(â1tâ«F (z + tâ1/2(x0 â y0) dX0(y0))Ã Ï0(z) dm(z) dX0(x0). (2.1.12)(b) There is a constant C2.1.7 such thatEXX0(Lt(1)2) â¤ C2.1.7(X0(1) t1â2Î»0 +X0(1)2 tâ2Î»0). (2.1.13)We note that the right hand side of (2.1.12) is equal to that of (2.1.5), and so wasoriginally computed in Proposition 1.5 of [69] as limÎ»ââEXX0(LÎ»t (Ï)). The fact that thesame formula gives the mean measure of Lt then follows from the L2 convergence of Lλt (Ï),as in Theorem 2.1.1.We first establish the existence of Lt, as well as its properties, under the measure N0,owing to the fact that the second moments of Lλt admit simpler formulas in this case. Inorder to prove the same for super-Brownian motion, we need to use the relationship betweensuper-Brownian motion under PXX0 and the canonical measure, which we now describe. Werecall that Nx is a Ï-finite measure such that Nx({Xt > 0}) = 2/t which describes theâlawâ of a single cluster of super-Brownian motion started at x; that is, the descendantsof a single ancestor at x. More precisely, super-Brownian motion is a superposition ofcanonical clusters; for a bounded, non-negative Borel function Ï : Râ R,EXX0 (exp (âXt(Ï))) = exp(ââ«â«1â eâµt(Ï)dNx0(µ) dX0(x0)). (2.1.14)This expression for the Laplace functional is in fact a consequence of a distributional equalitybetween super-Brownian motion under PXX0 and a Poisson point process of canonical clusters.For X0 â MF (R), let NX0(Â·) =â«Nx(Â·) dX0(x) and let ÎX0 be a Poisson point process onC([0,â),MF (R)) with intensity NX0 . We define a MF (R)-valued process (Xt : t â¥ 0) byXt(Â·) =ï£±ï£²ï£³â«µt(Â·) dÎX0(µ) if t > 0,X0(Â·) if t = 0.(2.1.15)By Theorem 4 of Section IV.3 of [57], (Xt : t â¥ 0) is a super-Brownian motion with initialmeasure X0. The âpointsâ of the point process ÎX0 are the clusters of X. For fixed t > 0,(2.1.15) leads toXt =âjâItµjt ,48where {µjt : j â It} are the points of a Poisson point process with finite intensity NX0(µt â· |µt > 0}. Let X¯0(Â·) = X0(Â·)/X0(1). Assuming our probability space is rich enough toallow us to choose random relabellings of these points, by the above we can writeXt =Nâi=1Xit , (2.1.16)where N is Poisson(2X0(1)/t) and, given N , {Xit : i = 1, . . . , N} are iid with distributionNX¯0(Xt â · |Xt > 0). We can and do condition on the values of the initial points ofthe clusters, denoted by x1, . . . , xN , which are iid points with distribution X¯0, in whichcase Xit has conditional distribution Nxi(Xt â · |Xt > 0). In order to prove the existenceand properties of Lt with respect to a super-Brownian motion Xt, we realize the super-Brownian motion as a point process and express Xt as above. Conditioning on N andapplying (2.1.16), we can write Lλt (Ï) asLλt (Ï) = Î»2Î»0â« [ Nâi=1Xi(t, x)]eâÎ»âNi=1Xi(t,x) Ï(x) dx.The almost sure existence of boundary local times corresponding to the canonical clustersallows us to take this limit quite easily and so establish that Lt exists under PXX0(i.e.Theorem 2.1.1). Furthermore, we obtain a conditional representation for Lt in terms of itsclusters; this allows us to transfer the properties of Lt under N0 to Lt under PXX0. Let Litdenote the boundary local time of Xit . In the statement that follows, we assume that wehave realized Xt using (2.1.16).Theorem 2.1.8 (Cluster decomposition). Let Xt be super-Brownian motion under PXX0and Lt be its boundary local time. Conditional on N , we havedLt(x) =Nâi=11(âj 6=iXj(t, x) = 0)dLit(x)= 1(X(t, x) = 0)Nâj=1dLit(x). (2.1.17)Remark. Given the nature of BZt, we expect this behaviour. In the cluster decomposition,each cluster has a boundary local time of its own. Since each is supported on the boundaryof its respective zero set, the local time Lt of Xt will be equal to the sum of cluster localtimes, except the boundary of the zero set of one cluster may be âswallowedâ by the supportof another, hence the indicator functions.49The idea of representing the boundary local time of Xt in terms of the boundary localtime of its clusters is not restricted to a super-Brownian motion and its comprising canonicalclusters. The following formulation of the same principle will be useful in Chapter 3. Recallthat a sum of independent super-Brownian motions is a super-Brownian motion.Theorem 2.1.9 (General cluster decomposition). Suppose X1, . . . ,Xn are independentsuper-Brownian motions with corresponding boundary local times Lit at time t > 0, fori = 1, . . . , n. Let X =âni=1Xi and let Lt be the boundary local time of Xt. ThendLt(x) =nâi=11(âj 6=iXj(t, x) = 0)dLit(x)= 1(X(t, x) = 0) nâi=1dLit(x).One example of superprocesses satisfying the above conditions follows from (III.1.3) of[83]. Let X0 â MF (R) and suppose that {A1, . . . , An} is a Borel partition of R. DefineXi as the contribution to X from ancestors at time 0 which are in Ai. (This makes Xia super-Brownian motion with initial measure X0(Â· â© Ai); a precise definition of Xi maybe given in terms of the historical process as in the above reference.) Then X =âNi=1Xisatisfies the conditions of the above theorem.Notations. We will make use of the common convention that C denotes any positive con-stant whose value is not important. The value of C may change line to line in a derivation;to bring attention to the fact that the constant has changed, we will sometimes label the newconstant C â². We write f â¼ g if limx f(x)/g(x) = 1, where the limit will be clear from con-text. As the reader has probably inferred, we will write µ > 0 when a measure has positivemass (that is, to indicate that µ(1) > 0). For an interval I â R, let C(I,R) denote thespace of continuous maps from I to R.Let St denote the semi-group of Brownian motion and pt the associated heat kernel(the Gaussian density of variance t). Let N (x0, Ï2) denote the law of a one-dimensionalGaussian with mean x0 and variance Ï2.Organization of the chapter. This chapter is organized as follows. Section 2.2 gives abrief overview of the theory of one-dimensional Ornstein-Uhlenbeck processes with Marko-vian killing. Our method relies on a change of variables which allow us to express certainquantities in terms of eigenvalue problems involving these processesâ generators.Section 2.3 describes fundamental background connecting the Laplace functional ofsuper-Brownian motion to a family of semi-linear PDEs. We also introduce the families V λand V Î»,Î»â², which play a key role in our analysis.In Section 2.4, we assume Theorem 2.1.4 (the L2-convergence result) and proceed to50prove our main results, including existence and properties of Lt and the cluster representa-tions. First we prove the existence of Lt under N0 (Theorem 2.1.1 for N0) and the formulaefor its first and second moments (Theorem 2.1.5). Next, we use the cluster decompositionto prove the existence of Lt under PXX0(Theorem 2.1.1 for PXX0) and its representation interms of clusters (Theorems 2.1.8 and 2.1.9). We then establish the upper bound on secondmoments of Lt under N0 (Proposition 2.1.6), which allows us to prove the remaining results,including Theorem 2.1.3, the dimension result.Section 2.5 contains the proof of Theorem 2.1.4, with the proof of one technical lemmagiven in Section 2.6.2.2 Killed Ornstein-Uhlenbeck processesAs above, we define the operator A by Af(x) = fâ²â²(x)2 â xfâ²(x)2 . The Markov process generatedby A is a one-dimensional Ornstein-Uhlenbeck process with mean zero. We denote thisprocess by Y , denote its law when started at x by P Yx with corresponding expectation EYx .For general initial conditions Y0 â¼ µ â M1(R) (the space of probability measures on R),we write its law as P Yµ . Y has a stationary measure, the unit variance Gaussian measure,m. When Y0 â¼ m, the process is reversible and can be defined for time values in R. Wewill denote the law of this stationary process on R by P Y .We now introduce the notions of killing and lifetime for the process (Yt : t â¥ 0). LetÏ â C+([ââ,â],R), the space of non-negative continuous functions with limits at Â±â.Such functions are also bounded. We will call this family of functions killing functions.Let AÏf(x) = Af(x) â f(x)Ï(x). AÏ is the generator of an Ornstein-Uhlenbeck processsubjected to Markovian killing at rate Ï(Yt). The lifetime of the killed process is ÏÏ =inf{t > 0 : â« t0 Ï(Ys) ds > e}, where e is an independent Exp(1) random variable. We recallthat the distribution of ÏÏ is given by (2.1.4).The generators A and AÏ correspond to strongly continuous contraction semigroups onL2(m). The following theorem is proved in [69], where it is stated as Theorem 2.3. Wenote that the statement of the result in that paper had a misprint when describing theconvergence of of the transition densities, which appeared in part (c). We have correctedthe statement, which is in part (b) of the following.Theorem 2.2.1. For Ï â C+([ââ,â],R), the following statements hold.(a) AÏ has a complete orthonormal family of C2 eigenfunctions {Ïn : n â¥ 1} in L2(m)satisfying AÏÏn = âÎ»nÏn, where 0 â¤ λ0 â¤ λ1 â¤ · · · â â. Furthermore, âÎ»0 is a simpleeigenvalue and Ï0 > 0.(b) For t > 0, the diffusion Y generated by AÏ has a jointly continuous transition density51qt(x, y) with respect to m, given byqt(x, y) =âân=0eâÎ»ntÏn(x)Ïn(y), (2.2.1)where the series converges in L2(m Ã m) and uniformly absolutely on sets of the form[Ç«,â)Ã [âÇ«â1, Ç«â1]2 for all Ç« > 0.(c) For 0 < δ < 12 , there exists a constant cδ > 0 such thatqt(x, y) â¤ cÎ´eâÎ»0teÎ´(x2+y2) for all t â¥ sâ(Î´), (2.2.2)where sâ(Î´) > 0 is the solution of2δ =eâsâ/2 â eâsâ1â eâsâ . (2.2.3)(d) Denote θ =â«Ï0 dm. For all t â¥ 0 and x â R,eÎ»0tPx(ÏÏ > t) = Î¸Ï0(x) + r(t, x), (2.2.4)where, for any δ > 0, there is a constant cδ > 0 such thatÏ0(x) â¤ cδeδx2 , (2.2.5)|r(t, x)| â¤ cÎ´eÎ´x2eâ(Î»1âÎ»0)t. (2.2.6)(e) As T ââ, Px(Y â Â·| ÏÏ > T )â P Y,âx weakly on C([0,â),R), where P Y,âx is the lawof the diffusion with the transition densityqËt(x, y) â¡ qt(x, y)Ï0(y)Ï0(x)eÎ»0t (2.2.7)with respect to m.The bounds in part (d) of the above easily imply the following estimates, which we willoften use. For 0 < δ < 1/2, there is a constant Cδ > 0 such thatP Yx (ÏÏ > t) â¤ CÎ´eÎ´x2eâÎ»0t âx â R, t > 0. (2.2.8)This implies that there is a constant C > 0 such thatP Ym (ÏÏ > t) â¤ CeâÎ»0t â t > 0. (2.2.9)52The following limit result is a simple consequence of the eigenfunction expansion for qt(x, y).Lemma 2.2.2. For all x, y â R,limtââ eλ0tqt(x, y) = Ï0(x)Ï0(y).The convergence is uniform on compact sets.Proof. For all t > 0 and x, y â R, from (2.2.1), we haveeλ0tqt(x, y) = Ï0(x)Ï0(y) +âân=1eâ(Î»nâÎ»0)tÏn(x)Ïn(y). (2.2.10)As we are taking t â â we can restrict to t â¥ 1, in which case the absolute value of thesum above is bounded above byeâ(Î»1âÎ»0)(tâ1)âân=1eâ(Î»nâÎ»0)|Ïn(x)Ïn(y)|.By Theorem 2.2.1(b) with t = 1, the series in the above is convergent, and the convergenceis uniform on compact sets. Part (a) of the same theorem states that âÎ»0 is a simpleeigenvalue. Hence Î»1â λ0 > 0 and the above vanishes as tââ; in fact, because the seriesconverges uniformly on compacts to a continuous limit, the above vanishes uniformly oncompacts as tââ, so (2.2.10) gives the result.It will be useful for us to study the distribution of the process Y when conditioned on sur-vival and its endpoint. Hereafter we assume that Y has killing function Ï â C+([ââ,â],R)and we denote its lifetime by Ï. For fixed T > 0 and z â R, consider the [0, T ]-indexedinhomogeneous Markov process taking values in R with transition density (with respect todm(y2))qËs,t(y1, y2) =qtâs(y1, y2) qTât(y2, z)qTâs(y1, z)(2.2.11)for 0 â¤ s < t < T . (The kernels are degenerate when t = T , since YT = z.) Below we verifythat the finite dimensional distributions defined by this transition kernel have an extensionto a (necessarily) unique law on C([0, T ],R), which we denote by P Yx (Â· | Ï > T, YT = z)when the initial point is x â R, and show that it gives an explicit version of the suggestedregular conditional distribution for all z â R. We then establish that for fixed S > 0,P Yx (Y |[0,S] â · | Ï > T, YT = z) converges weakly to PY,âx (Y |[0,S] â Â·) as T â â for allz â R.Lemma 2.2.3. (a) Let x â R and T > 0. For all z â R, the finite dimensional distributionsdescribed in (2.2.11), with initial value x, have a unique extension to C([0, T ],R). The53resulting laws P Yx (Â· | Ï > T, YT = z) are continuous in z and define a regular conditionalprobability for Y |[0,T ] under PYx conditioned on YT .(b) Let x, z â R, S > 0 be fixed. Then Px(Y |[0,S] â · | Ï > T, YT = z) converges weakly onC([0, S],R) to P Y,âx (Y |[0,S] â Â·) as T ââ.(c) For all S,K > 0,{Px(Y |[0,S] â · | Ï > T, YT = z) : |x|, |z| â¤ K,T â¥ S}is tight onC([0, S],R).Before proving the lemma, we make an observation concerning time reversals of Yunder P Yx (Â· | Ï > T, YT = z). For T > 0 and t â [0, T ], define YËt = YTât. Let x, z â R. For0 < t1 < t2 < T and Ï1, Ï2 bounded Borel functions, we haveEYx (Ï1(YËt1)Ï2(YËt2)â£â£ Ï > T, YT = z)=1qT (x, z)â«â«Ï1(y1)Ï2(y2) qTât2(x, y2) qt2ât1(y2, y1) qt1(y1, z) dm(y1) dm(y2)= EYz (Ï1(Yt1)Ï2(Yt2)â£â£ Ï > T, YT = x),where the last equality uses qt(x, y) = qt(y, x). The above equality of distributions can beextended to general finite dimensional distributions. Because the extension of the finitedimensional distributions to a law on C([0, T ],R) (i.e. from Lemma 2.2.3(a)) is unique, wetherefore have that for all x, z â R,P Yx (YË |[0,T ] â · | Ï > T, YT = z) = PYz (Y |[0,T ] â · | Ï > T, YT = x). (2.2.12)As a last note, we will sometimes denote the law P Yx (Â· | Ï > T, YT = z) simply by P Yx (Â· |YT =z) when it is clear from context that we are working with the killed process.Proof of Lemma 2.2.3. Let x, z â R and T > 0. We define a distribution, which we denoteby P Yx (Â· | Ï > T, YT = z), on finite (time-indexed) collections of random variables, whichdescribes the finite dimensional distributions (FDDs) of the inhomogeneous Markov processwith transition density (2.2.11). For 0 = t0 < t1 < · · · < tn < T and bounded, continuousfunctions Ï1, . . . , Ïn, we define the n-dimensional FDD of (Yt1 , . . . , Ytn) under PYx (Â· | Ï >T, YT = z) asEYx( nâi=1Ïi(Yti)â£â£â£â£ Ï > T, YT = z)=1qT (x, z)â« [ nâi=1Ïi(yi)qtnâtnâ1(ynâ1, yn)]qTâtn(yn, z)nâi=1dm(yi), (2.2.13)where we use the convention y0 = x. We note that (2.2.13) also defines the FDDs of a54regular conditional distribution of (Yt : t â [0, T ]) under P Yx conditioned on YT = z (whichis why we have used this notation). Thus when we have established that these laws extendto a probability on C([0, T ],R), we will have explicitly constructed a version of the regularconditional distribution.To prove that P Yx (Â· | Ï > T, YT = z) extends to a probability on C([0, T ],R), we willestablish a tightness criterion. We consider the fourth moments of increments of Y . Let0 < s < t < T . Expanding using (2.2.13), we haveEYx ((Yt â Ys)4 | Ï > T, YT = z)=1qT (x, z)â«â«(y2 â y1)4qs(x, y1) qtâs(y1, y2) qTât(y2, z) dm(y1) dm(y2). (2.2.14)We now collect some elementary bounds and inequalities which will allow us to obtain auseful upper bound for the above. First, we note that while qt(x, y) is a transition densitywith respect to m, it will sometimes be useful to express it as a density with respect to theLebesgue measure. Since p1(Â·) is the density of m, we haveqt(x, y) dm(y) = qt(x, y) p1(y) dy. (2.2.15)We will use a comparison with an un-killed Ornstein-Uhlenbeck process. The transitionkernel of a standard Ornstein-Uhlenbeck process is described by, for 0 â¤ s < t,(Yt â Ys |Ys = y) â¼ N (eâ(tâs)/2y, 1â eâ(tâs)).Let kt(x, y) denote the transition density of an un-killed Ornstein-Uhlenbeck process withrespect to Lebesgue measure. Then for x, y â R and t > 0,kt(x, y) =(2Ï)â1/2â1â eât exp(â(y â eât/2x)2/ 2(1 â eât)) . (2.2.16)The transition densities of the killed Ornstein-Uhlenbeck process are bounded above bythose of the un-killed process. This implies that(i) qt(x, y) dm(y) â¤ kt(x, y) dy, (ii) qt(x, y) p1(y) â¤ kt(x, y). (2.2.17)It is easy to establish from (2.2.16) that there is a constant c > 0 such thatkt(x, y) â¤ cpt(y â xeât/2) for all t â¤ 2 and x, y â R. (2.2.18)where we recall that pt(Â·) is the Gaussian density of variance t. Let K > 0. From (2.2.17)(ii)55and (2.2.18) it follows that there is a constant C1(K) such thatqTât(y2, z) â¤ kTât(y2, z)p1(z)â1 â¤ C1(K)âT â T â² â y2 â R, z â [âK,K], and t â¤ Tâ² < T.(2.2.19)Next, we note that it holds by elementary formulas for moments of Gaussians that there isa constant c > 0 such thatâ«(y2 â y1)4pt(y2) dy2 â¤ c (t2 + |y1|4) â y1 â R, t > 0. (2.2.20)Finally, observe that qT (Â·, Â·) is bounded below by the transition density of Y with constantkilling function âÏââ. Thus for all K > 0 and M â¥ 1, from (2.2.15) we haveqT (x, z) â¥ eââÏââTkT (x, z)p1(z)â1 â¥ δ(K,M) âx, z â [âK,K], T â [Mâ1,M ] (2.2.21)for a sufficiently small constant δ(K,M) > 0.Let 0 < T â² < T and suppose that 0 < s < t â¤ T â² such that tâs â¤ 1. Let K > 0 and sup-pose that x, z â [âK,K]. Using (2.2.17)(i) to bound qs(x, y1)dm(y1) and qtâs(y1, y2)dm(y2),and (2.2.19) to bound qTât(y2, z), from (2.2.14) we obtain thatEYx ((Yt â Ys)4 | Ï > T, YT = z)â¤ C1(K) (T â Tâ²)â1/2qT (x, z)â«ks(x, y1)[ â«(y2 â y1)4ktâs(y1, y2)dy2]dy1â¤ C1(K) (T â Tâ²)â1/2qT (x, z)â«ks(x, y1)[ â«c(y2 â y1)4ptâs(y2 â eâ(tâs)/2y1)dy2]dy1,(2.2.22)where the second inequality uses (2.2.18). Changing variables and applying (2.2.20), weobtain thatâ«c(y2 â y1)4ptâs(y2 â eâ(tâs)/2y1)dy2 â¤ C(|y1|4(1â eâ(tâs)/2)4 + (tâ s)2)â¤ C(1 + |y1|4)(tâ s)2, (2.2.23)where in the second inequality we have 1â eâx â¤ x for x â¥ 0 and (tâ s)4 â¤ (tâ s)2 (sincetâ s â¤ 1). Substituting this into (2.2.22), we obtain thatEYx ((YtâYs)4 |YT = z) â¤C1(K) (T â T â²)â1/2qT (x, z)(tâs)2â«C ks(x, y1) (|y1|4+1) dy1. (2.2.24)Recall that we have assumed x, z â [âK,K]. By (2.2.16) it is clear that for K > 0,56the integral is bounded above by some constant C2(K) > 0 for all x â [âK,K] and s > 0.Using this along with (2.2.21), with a choice of M â¥ 1 for which T â [Mâ1,M ], from theabove we deduce the following:For all x, z â [âK,K], 0 < s < t â¤ T â² such that tâ s â¤ 1,EYx ((Yt â Ys)4 |YT = z) â¤ C2(K) Î´(K,M)â1 C1(K) (T â T â²)â1/2 Ã (tâ s)2. (2.2.25)Let T â² = 2T/3. Hereafter we consider increments of size at most 1 â§ T/3. We havethat (2.2.25) holds for all 0 < s < t â¤ 2T/3 such that t â s â¤ 1 â§ T/3 with constantC2(K)Î´(K,M)â1C1(K)(T/3)â1/2. It remains to show that it also holds on [2T/3, T ]. Todo so, we make use of reversibility. Suppose T/3 â¤ s < t < T . ThenEYx ((Yt â Ys)4â£â£YT = z)=1qT (x, z)â«â«(y2 â y1)4 qs(x, y1) qtâs(y1, y2) qTât(y2, z) dm(y1) dm(y2)= EYz ((YTâs â YTât)4â£â£YT = x), (2.2.26)where the last equality uses qt(x, y) = qt(y, x) (a consequence of (2.2.1)) and (2.2.13). Since0 < T â t < T â s â¤ 2T/3, by (2.2.25) and (2.2.26) we have that for all x, z â [âK,K] andT/3 â¤ s < t < T such that tâ s â¤ 1 â§ T/3,EYx ((Yt â Ys)4 | Ï > T, YT = z) = C2(K) δ(K,T )â1 C1(K) (T/3)â1/2 Ã (tâ s)2.Combined with the previous statement that this holds for all 0 < s < t â¤ 2T/3, we havethatFor all x, z â [âK,K], 0 < s < t < T such that tâ s â¤ 1 â§ T/3,EYx ((Yt â Ys)4 |YT = z) â¤ C2(K) Î´(K,M)â1 C1(K) (T/3)â1/2 Ã (tâ s)2. (2.2.27)The above proof can be easily modified to obtain the same bound (with a potential changeto the constant) for increments in which s = 0 or t = T , and we omit it. Thus by (2.2.27)and the Kolmogorov Continuity Theorem, P Yx (Â· | Ï > T, YT = z) has a unique extension to aprobability on C([0, T ],R), also denoted by P Yx (Â· | Ï > T, YT = z). As we noted earlier, thisgives an explicit construction of the regular conditional distribution (Yt : t â [0, T ]) underP Yx given Ï > T and YT = z, Additionally, suppose that zn â z and that zn â [âK,K]for all n â¥ 1. From (2.2.27), {P Yx (Â·| Ï > T, YT = zn) : n â¥ 1} is tight. It is clear from(2.2.13) and continuity of qt(Â·, Â·) that the FDDs of P Yx (Â· | Ï > T, YT = zn) converge to those ofP Yx (Â· | Ï > T, YT = z). Thus the aforementioned tightness proves that P Yx (Â· | Ï > T, YT = zn)57converges to P Yx (Â· | Ï > T, YT = z) as a law on C([0, T ],R). Thus we have proved part (a).Before proving (b), we note the following consequence of (2.2.27) and its proof. LetS,K > 0 and fix M > 1 such that S â [Mâ1,M ]. By considering increments of (Ys : s â[0, S]) but allowing the time T at which we condition YT = z to take values in [S,M ], wehave that{P Yx (Y |[0,S] â · | Ï > T, YT = z) : |x|, |z| â¤ K,T â [S,M ]} is tight. (2.2.28)Next we turn to part (b). Fix S > 0 and x, z â R. We now check that the FDDs of(Ys : s â [0, S]) under P Yx (Â· | Ï > T, YT = z) converge to those of (Ys : s â [0, S]) under P Y,âas T â â. Let 0 < t1 < t2 â¤ S and let Ï1 and Ï2 be bounded and continuous functions.Then from (2.2.13), we haveEYx (Ï1(Yt1)Ï2(Yt2)â£â£ Ï > T, YT = z)=1qT (x, z)â«â«Ï1(y1)Ï2(y2) qt1(x, y1) qt2ât1(y1, y2) qTât2(y2, z) dm(y1) dm(y2)=eλ0t2eλ0T qT (x, z)â«â«Ï1(y1)Ï2(y2) qt1(x, y1) qt2ât1(y1, y2)Ã eÎ»0(Tât2)qTât2(y2, z) dm(y1) dm(y2). (2.2.29)By Lemma 2.2.2, we havelimTââeÎ»0(Tât)qTât(y2, z) = Ï0(y2)Ï0(z), limTââeÎ»0T qT (x, z) = Ï0(x)Ï0(z). (2.2.30)Moreover, applying (2.2.2) with δ = 1/8, we have thateÎ»0(Tât)qTât(y2, z) â¤ cey22/8+z2/8 â y2, z â R, t â (0, S] and T â¥ S + sâ(1/8), (2.2.31)where sâ(1/8) is as in (2.2.3). Using (2.2.31) (replacing t with t2) and (2.2.17)(i) we obtainthe following bound for the integrand in (2.2.29):|Ï1(Yt1)Ï2(Yt2) qt1(x, y1) qt2ât1(y1, y2) eÎ»0(Tât2)qTât2(y2, z)| dm(y1) dm(y2)â¤ cez2/8âÏ1âââÏ2ââey22/8kt1(x, y1)kt2ât1(y1, y2) dy1 dy2for all T â¥ S+sâ(1/8). By (2.2.16), kt1(x, y1) and kt2ât1(y1, y2) are Gaussians with varianceat most 1, and so a short argument shows that the above quantity is integrable. This allows58us to use Dominated Convergence in (2.2.29), so by (2.2.30) we havelimTââEx(Ï1(Yt1)Ï2(Yt2)â£â£YT = z)=eÎ»0t2Ï0(x)Ï0(z)â«â«Ï1(y1)Ï2(y2) qt1(x, y1) qt2ât1(y1, y2)Ï0(y2)Ï0(z)dm(y1)dm(y2)=â«â«Ï1(y1)Ï2(y2)[eÎ»0t1qt1(x, y1)Ï0(y1)Ï0(x)] [eÎ»0(t2ât1)qt2ât1(y1, y2)Ï0(y1)Ï0(y2)]dm(y1)dm(y2)=â«â«Ï1(y1)Ï2(y2) qËt1(x, y1) qËt2ât1(y1, y2)dm(y1)dm(y2)= EY,âx (Ï1(Yt1)Ï2(Yt2)).The above argument can be easily generalized to n-fold FDDs for all n â¥ 2, (the δ = 1/8in (2.2.31) can be reduced to handle larger n) and thus we have the desired convergence ofthe FDDs as T â â. In order to obtain weak convergence of the laws on C([0, S],R), weneed tightness of the distributions as T ââ. To prove that the distributions are tight wewill analyse the fourth moments of increments, as in (2.2.14), but first we obtain one morebound. We note that by Lemma 2.2.2 and joint continuity of (T, (x, y))â qT (x, y), it holdsthat for all K > 0,eλ0T qT (x, z) â¥ δ(K) > 0 âx, z â [âK,K], T â¥ 1 (2.2.32)for sufficiently small δ(K) > 0. Let K > 0 and x, z â [âK,K]. In (2.2.14), we boundqTât(y2, z) above using (2.2.31) and bound the other transition densities using (2.2.17),which givesEYx ((Yt â Ys)4 | Ï > T, YT = z)â¤ eλ0t+z2/8eλ0T qT (x, z)â«ks(x, y1)[ â«(y2 â y1)4 ktâs(y1, y2) ey22/8dy2]dy1â¤ eλ0S+K2/8 Î´(K)â1â«ks(x, y1)ey21/4[ â«c (y â y1(1â eâ(tâs)/2))4ptâs(y) ey2/4 dy]dy1â¤ eλ0S+K2/8 Î´(K)â1â«câ² ks(x, y1)ey21/4[ â«câ² (y â y1(1â eâ(tâs)/2))4 p2(tâs)(y) dy]dy1for all T â¥ S + sâ(1/8). In the second inequality we have used (2.2.32) as well as (2.2.18)and a change of variables. The third follows from a short calculation and the fact thattâ s â¤ 1. Applying (2.2.20) to the above and arguing as in (2.2.23), we obtain that, for all59T â¥ S + sâ(1/8),EYx ((Yt â Ys)4 | Ï > T, YT = z)â¤ eλ0S+K2/8 Î´(K)â1 (tâ s)2Câ«ks(x, y1) ey21/4 (1 + |y1|4) dy1,â¤ C3(S,K) (tâ s)2 âx, z â [âK,K], 0 â¤ s < t â¤ S such that tâ s â¤ 1, (2.2.33)for a constant C3(S,K) > 0, where to see that the integral is bounded uniformly for|x| â¤ K, we use the fact, from (2.2.16), that ks(x, y1) is Gaussian with mean of absolutevalue bounded above by |x| and variance less than 1. The fact that (2.2.33) holds for allT â¥ S + sâ(1/8) implies that the laws P Yx (Y |[0,S] â · | Ï > T, YT = z) are tight as T â â.Combined with the convergence of the FDDs to those of P Y,âx , this proves (b).Observe that (2.2.33) proves part (c) if we restrict to T â¥ S + sâ(1/8). If we chooseM â¥ 1 such that Mâ1 < S < S + sâ(1/8) < M , then (2.2.28) gives tightness of the lawsfor T â [S, S + sâ(1/8)]. Combining these two cases gives the desired tightness and proves(c).2.3 Some non-linear PDELet Bb+(R) denote the space of bounded, non-negative Borel functions. Recall that Stdenotes the semigroup of Brownian motion. By Theorem III.5 of [57], for Ï â Bb+(R), thereexists a unique non-negative solution, denoted V Ït (x), to the evolution equationVt = St(Ï)â 12(â« t0Stâs(V 2s ) ds), (2.3.1)such thatEXX0(eâXt(Ï))= eâX0(VÏt ) (2.3.2)for all X0 â MF (R). Applying the above with X0 = δx, (2.1.14) givesNx(1â eâXt(Ï)) = V Ït (x). (2.3.3)We are interested in the case when the initial data is a measure, and also in the differentialform of the equation. The integral equation (2.3.1) has a corresponding PDE, which is thefollowing:âVât=12â2Vâx2â V22for (t, x) â (0,â)Ã R, Vt â Ï as t â 0. (2.3.4)60In [5], this equation was shown to have a unique C1,2 solution when Ï â MF (R), whereVt â Ï is understood as weak convergence of measures. That is, we identify the functionVt with the measure Vt(x)dx, which converges weakly to Ï. By Lemma 2.1 of [70], thesolution of (2.3.4) is also the unique solution to (2.3.1). We denote the unique solutionto (2.3.1) and (2.3.4) by V Ït . Part (d) of the same lemma establishes that if Ïn â Ïweakly as n â â, then V Ïnt (x) â V Ït (x) for all t > 0, x â R. We note from (2.3.1) thatV Ïnt â¤ StÏn â¤ ctâ1/2Ïn(R). Using this and the fact that Xt has a bounded, continuousdensity, if we approximate measures by functions in Bb+(R), we can take bounded limits in(2.3.2) and (2.3.3) to establish that (2.3.2) and (2.3.3) hold for V Ït when Ï â MF (R).Notation. As Xt is absolutely continuous, when Ï â MF (R) we interpret Xt(Ï) asâ«X(t, x) dÏ(x).We now state some useful properties of solutions to (2.3.4). For a proof, see Lemma 2.6in [70].Proposition 2.3.1. Let Ï,Ï â MF (R).(a) (Monotonicity) If Ï â¤ Ï, then 0 â¤ V Ït â¤ V Ït for all t > 0.(b) (Sub-additivity) V Ï+Ït â¤ V Ït + V Ït for all t > 0.Next we fix Ï = λδx â MF (R) for λ > 0, so that Xt(Ï) = λX(t, x). Denote by V λt theunique, non-negative C2,1 solution to the initial value problemâVât=12â2Vâx2â V22for (t, x) â (0,â) Ã R, Vt â λδ0 weakly as t â 0. (2.3.5)This family was originally studied in [41]. It is an exercise to use (2.3.5) or the scalingproperties of super-Brownian motion to show that V λt (x) satisfies the following space-timescaling relationship. For λ, r > 0, we haveV λrt (x) = λ2V rλ2t(λx). (2.3.6)By translation invariance in the initial conditions of (2.3.5), and by (2.3.2) and (2.3.3) wehaveEXÎ´0(eâÎ»X(t,x))= eâVÎ»t (x), (2.3.7)N0(1â eâÎ»X(t,x)) = V λt (x) (2.3.8)for all x â R and t > 0. It is clear from (2.3.7) that V λt increases to a limit as Î»ââ. In thePDE literature this was established in [41], where it was shown that V λ converges locallyuniformly as Î»â â to a function Vât on (0,â) Ã R. Heuristically, Vât is the solution of61(2.3.5) when λ = +â. Rigorously, it is the unique solution to the following problem:âVât=12â2Vâx2â V22for (t, x) â (0,â) Ã R,limtâ0Vt(x) = 0 âx 6= 0, limtâ0â«BÇ«Vt(x) dx = +â â Ç« > 0, (2.3.9)where BÇ« = B(0, Ç«), the ball with radius Ç« centered at the origin. Vât was introduced andshown to solve (2.3.9) in [6]; uniqueness of the solution is a consequence of Theorem 3.5 of[64]. Taking Î»ââ in (2.3.8), we see that Vât satisfiesVât (x) = N0({X(t, x) > 0}). (2.3.10)We recall that (see Theorem II.7.2 of [83])N0({Xt > 0}) = 2/t. (2.3.11)Thus (2.3.10) implies thatVât (x) â¤ 2/t âx. (2.3.12)Taking λ2 = 1/t and letting r ââ in (2.3.6), one obtains that Vât (x) = tâ1Vâ1 (tâ1/2x).Definition. Define F : Râ R+ byF (x) := Vâ1 (x). (2.3.13)It follows that Vât (x) = tâ1F (tâ1/2x). It was shown in [6] that F is the solution to anODE problem. (In fact, their PDEs and ODEs have different (constant) coefficients, butSection 3 of [69] shows that F is a rescaled version of the function they study.) F is theunique solution of(i) F â²â²(x) + xF â²(x) + F (x)(2 â F (x)) = 0(ii) F > 0, F â C2(R) (2.3.14)(iii) F â²(0) = 0, F (x) â¼ c1|x|eâx2/2 as |x| â âfor some c1 > 0. We recall that f(x) â¼ h(x) means f(x)/h(x) â 1 as x â â. This Fis the function we discussed in the introduction, for which âÎ»0 is the lead eigenvalue ofthe operator AF . In particular, by evaluating (2.3.10) at t = 1 we can recover (2.1.3), ourpreliminary definition.As part of the proof of Theorem 2.A, the authors of [69] computed the rate of convergence62of V λt to Vât . In particular, Proposition 4.6 of that reference states thatsupxâR[Vât (x)â V λt (x)]â¤ Ctâ1/2âÎ»0Î»1â2Î»0 (2.3.15)for some constant C. (This is closely connected to (2.1.5).) A similar lower bound withthe same power of λ is established in the same proposition, although in this case one mustbe careful when t is close to zero. We will make frequent use of (2.3.15) in this chapter tobound error terms arising when we make approximations to obtain an eigenvalue problem.Let Y be an Ornstein-Uhlenbeck process. We define ZT (Y ) asZT (Y ) = exp(â« T0F (Ys)â V es/21 (Ys) ds). (2.3.16)Since V es/21 â Vâ1 = F as s ââ, the integrand converges to zero as s â â. As ZT (Y ) isincreasing in T , we can define Zâ(Y ) := limTââZT (Y ). By (2.3.15), we can easily deducethat the (monotone) limitZâ(Y ) := limTââZT (Y ) (2.3.17)exists and is finite, and that moreover there is a constant CZ > 0 such that, uniformly forall Y ,ZT (Y ) â¤ Zâ(Y ) â¤ CZ <â âT > 0. (2.3.18)Finally, we introduce another family of solutions to (2.3.4), which arise when we computesecond moments of Lλt ; we will evaluate expressions that involve the density at two pointsx1, x2 â R. Let V (Î»,Î»â²),(x1,x2)t denote VÏt when Ï = λδx1 + Î»â²Î´x2 â MF (R), so that, by(2.3.3),V(Î»,Î»â²),(x1,x2)t (y) = Ny(1â eâÎ»X(t,x1)âÎ»â²X(t,x2)). (2.3.19)When evaluating this function at 0, we will denote it by V Î»,Î»â²t (x1, x2) = V(Î»,Î»â²),(x1,x2)t (0).In other words,V Î»,Î»â²(x1, x2) = N0(1â eâÎ»X(t,x1)âÎ»â²X(t,x2)). (2.3.20)By (2.3.19), (2.3.20) and translation invariance of the canonical measure, these familiessatisfyV(Î»,Î»â²),(x1,x2)t (y) = V(Î»,Î»â²),(x1ây,x2ây)t (0) = VÎ»,Î»â²t (x1 â y, x2 â y). (2.3.21)Lastly, as can be readily seen from (2.3.20) and the symmetry of the canonical measure,V Î»,Î»â²t (x1, x2) = VÎ»,Î»â²t (âx1,âx2) (2.3.22)63for all x1, x2 â R. This family also satisfies a following scaling relationship which can bederived from studying the associated PDE directly. In particular,V rÎ»,cÎ»â²t (x1, x2) = Î»2Vr,cÎ»â²/Î»Î»2t(Î»x1, λx2) = (Î»â²)2V rÎ»/Î»â²,c(Î»â²)2t(Î»â²x1, Î»â²x2), (2.3.23)for all λ, Î»â², r, c > 0 and x1, x2 â R. Taking limits and applying bounded convergencein (2.3.2), we see that V Î»,Î»â²t (x1, x2) has a monotone limit as λ, Î»â² â â (by Proposi-tion 2.3.1(a)). We denote this limit Vâ,ât (x1, x2). In agreement with our previous notationwe define the following.Definition. We define F2 : R2 â R+ byF2(x1, x2) := Vâ,â1 (x1, x2). (2.3.24)By taking the limit as λ, Î»â² â â in (2.3.20) (and in (2.3.2) with Ï = λδx1 + Î»â²Î´x2) weobtain thatVâ,ât (x1, x2) = N0({X(t, x1) > 0} âª {X(t, x2) > 0}) = â logPXδ0 (X(t, x1) = X(t, x2) = 0).(2.3.25)We conclude by stating a version of (2.3.15) for the functions V Î»,Î»â²t .Lemma 2.3.2. There is a positive constant C such that for all t, λ, Î»â² > 0,supx1,x2âR[Vâ,ât (x1, x2)â V Î»,Î»â²t (x1, x2)]â¤ Ctâ1/2âÎ»0[Î»1â2Î»0 + Î»â²1â2Î»0].Proof. Let x1, x2 â R and t, λ, Î»â² > 0. We writeVâ,ât (x1, x2)â V Î»,Î»â²t (x1, x2)=[Vâ,ât (x1, x2)â V Î»,ât (x1, x2)]+[V Î»,ât (x1, x2)â V Î»,Î»â²t (x1, x2)]. (2.3.26)Using (2.3.25) and (2.3.20) and taking Î»â² â â in the latter, it follows that the first termin the above is equal toN0 (1â 1(X(t, x1) = X(t, x2) = 0))â N0(1â eâÎ»X(t,x1)1(X(t, x2) = 0))= N0(1(X(t, x2) = 0)(eâÎ»X(t,x1) â 1(X(t, x1) = 0)))â¤ N0(eâÎ»X(t,x1) â 1(X(t, x1) = 0))= Vât (x1)â V λt (x1)â¤ Ctâ1/2âÎ»0Î»1â2Î»0 ,64where the second last line follows from (2.3.10) and (2.3.8), and the final inequality isby (2.3.15). We use similar reasoning to bound the second term of (2.3.26) by the sameexpression with Î»â² replacing λ, which gives the desired result.2.4 Existence and properties of LtAs stated in the introduction, our method first establishes the existence and properties of Ltunder N0 and then uses the cluster decomposition to establish them under PXX0. The mainingredient in the proof of Theorem 2.1.1 is the convergence of second moments of Lλt (Ï) asλ â â. For a bounded Borel function Ï, we show that N0(Lλt (Ï)2) converges as λ â â.In fact, we prove convergence of second moments of general functions of two variables. Forh : R2 â R we recall the notation(Lλt Ã Lλt )(h) =â«h(x, y) dLλt (x) dLλt (y).Lλt (Ï)2 is easily recovered by taking h(x, y) = Ï(x)Ï(y). The following result is theworkhorse of this chapter.Theorem 2.1.4. There exists a constant C2.1.4 > 0 and continuous function Ï : RÃRâ(0, 1] such that for bounded Borel h : R2 â R,limÎ»,Î»â²ââN0((LÎ»t Ã LÎ»â²t )(h))= C22.1.4â« t0(tâ s)â2Î»0[ â«â«EB0(h(âtâ s z1 +Bs,âtâ s z2 +Bs))Ã exp(ââ« s0Vâ,âtâu (âtâ s z1 +Bs âBu,âtâ s z2 +Bs âBu) du)Ã Ï(z1, z2)Ï0(z1)Ï0(z2) dm(z1) dm(z2)]ds.Corollary 2.4.1. For a bounded Borel function Ï, Lλt (Ï) converges in L2(N0) as Î»ââ.Proof. Since L2(N0) is complete, it is enough to show that {Lλt (Ï)}Î»>0 is Cauchy in L2(N0).For λ, Î»â² > 0, we haveN0((Lλt (Ï)â LÎ»â²t (Ï))2) = N0((Lλt (Ï)2) + N0((LÎ»â²t (Ï)2)â 2N0(Lλt (Ï)LÎ»â²t (Ï)).By Theorem 2.1.4, this converges to 0 as λ, Î»â² ââ.The proof of Theorem 2.1.4 is long and technical. We defer it to Section 2.5, which isdevoted to its proof. For now, we assume the result and use it to establish our other main65results, the first being the existence of Lt under N0.Proof of Theorem 2.1.1 for N0. Fix t > 0. Because Xt = 0 implies that Lλt = 0 for allλ > 0, without loss of generality we can work under the finite measure N0(Â·â©{Xt > 0}). ByCorollary 2.4.1, for a bounded continuous function Ï, there exists a random variable l(t, Ï)such that Lλt (Ï)â l(t, Ï) in L2(N0) as Î»ââ. It follows that Lλt (Ï)â l(t, Ï) in measure.We will now establish that there exists a unique random measure Lt such that the randomvariable l(t, Ï) is the integral of Ï with respect to a random measure Lt, i.e. l(t, Ï) = Lt(Ï)for all continuous and bounded functions Ï.We need to establish that the measures {Lλt : λ > 0} are tight N0-almost surely. Tosee that this is true, we recall that X(t, Â·) is compactly supported N0-a.s., (see CorollaryIII.1.4 of [83] for the result under PXδ0 ; condition the cluster representation on N = 1 to getit for N0) and hence the mass of Xt is contained in a ball B(0, R) for some R = R(Ï) > 0.Since Lλt (A) = Î»2Î»0â«AX(t, x)eâÎ»X(t,x) dx, this implies that the mass of Lλt is contained inB(0, R) for all λ > 0, which implies that {Lλt (Ï) : λ > 0} is tight.Let {Ïn}ân=1 be a countable determining class for MF (R) consisting of bounded, con-tinuous functions. We choose Ï1 = 1. L1-boundedness of the total mass and tightness aresufficient conditions for a family in MF (R) (with the weak topology) to be relatively com-pact. By Corollary 2.4.1, {Lλt (1) : λ > 0} is L2(N0)-bounded, and hence L1(N0)-bounded,and so from the above we see that{Lλt : λ > 0} is relatively compact N0-a.s.As we have noted, Lλt (Ïn)â l(t, Ïn) in measure as Î»ââ. Using the fact that conver-gence in measure implies almost sure convergence along a subsequence, we can iterativelydefine subsequences and take a diagonal subsequence {Î»m}âm=1 which satisfiesLλmt (Ïn)â l(t, Ïn) as mââ for all n â¥ 1 N0-a.s. (2.4.1)As shown above, {Lλmt }âm=1 is relatively compact N0-almost surely. Combined with (2.4.1),this means that for N0-a.a. Ï we have the above convergence for all n â¥ 1 and relativecompactness of the measures {Lλmt }âm=1. Choose such an Ï. By relative compactness of{Lλt }λ>0, any subsequence of {Î»m}âm=1 admits a further sequence along which the measuresconverge in the weak topology. It remains to show that all subsequential limits coincide.Suppose Lt(Ï) and Lâ²t(Ï) are two such limit measures. Since Ï has been chosen so that(2.4.1) holds, we have that Lt(Ï)(Ïn) = Lâ²t(Ï)(Ïn) for all n. Since the family {Ïn}nâ¥1 are adetermining class, this implies that Lt(Ï) = Lâ²t(Ï). Hence all subsequences admit a further66subsequence with the same limit Lt(Ï) in the weak topology. Since the weak topology onMF (R) is metrizable, the âevery subsequence admits a further converging subsequenceâcriterion for convergence applies, and we have Lλmt (Ï) converges to Lt(Ï) â MF (R) asmââ. This gives the almost sure convergence along {Î»m}âm=1.We now check that Lλt â Lt in measure as λ â â. First note that we can restrictto the finite measure N0(Â· â© {Xt > 0}), since Lλt = Lt = 0 for all λ > 0 on {Xt = 0}.Let d(µ, ν) be a metric which metrizes the weak topology on MF (R), e.g. the Wassersteinmetric (see p. 48 of [82]). If Lλt did not converge to Lt in measure, then there would be asequence {Î»¯k}âk=1 and Ç«, δ > 0 such that N0({d(Lλ¯kt , Lt) > Ç«} â© {Xt > 0}) > δ for all k â¥ 1.However, using the previous argument we can obtain a subsequence on which the measuresconverge to Lt N0(Â· â© {Xt > 0})-a.s., which, because N0(Â· â© {Xt > 0}) is a finite measure,contradicts the previous statement. Hence we must have that Lλt â Lt in measure.Next, we observe that for continuous and bounded Ï, Lt(Ï) = l(t, Ï). To see this, recallthat Lλmt (Ï) converges to l(t, Ï) in L2(N0). As we have just shown that limmââ Lλmt (Ï) =Lt(Ï) N0-a.s, it must hold that Lt(Ï) = l(t, Ï). This implies that Lλt (Ï)â Lt(Ï) in L2(N0)by Corollary 2.4.1.Finally, we verify that Lt is supported on BZt. We fix Ï outside of a null set suchthat Lλmt â Lt in MF (R) as m â â. For an open set U , Lt(U) â¤ lim infmââLÎ»mt (U)(a consequence of the Portmanteau theorem). From (2.1.6), we have Lλmt (Zt) = 0 forall m â¥ 1, which implies that Lt(int(Zt)) = 0. Moreover, X(t, x) > 0 implies thatλ2λ0m X(t, x)eâÎ»mX(t,x) â 0 as m â â, so for Ç« > 0, Lt({x : X(t, x) > Ç«}) = 0, andhence Lt(Zct ) = 0. Since Lt(int(Zt) âª Zct ) = 0, we must have supp(Lt) â BZt.Proof of Theorem 2.1.5. To prove (b), by Theorem 2.1.4 it is enough to show that N0((LtÃLt)(h)) = limnââN0((LÎ»nt Ã Lλnt )(h)) for a sequence λn â â, which we choose to bethe sequence from Theorem 2.1.1 on which Lλnt â Lt almost surely. Because Lt = 0when Xt = 0, we can work on the probability measure N0(Â· |Xt > 0). For bounded andcontinuous h : R2 â R, |(Lλnt Ã Lλnt )(h)| â¤ âhââLÎ»nt (1)2. By Theorem 2.1.1, Lλnt (1)converges in probability and in L2(N0(Â· |Xt > 0)) to Lt(1), which implies that Lλnt (1)2 andhence (Lλnt Ã Lλnt )(h) are uniformly integrable (see, e.g. Theorem 4.6.3 of [19]). We cantherefore exchange limit and expectation, givingN0( limnââ(LÎ»nt Ã Lλnt )(h)) = limnââN0((LÎ»nt Ã Lλnt )(h)).Since Lλnt â Lt inMF (R) and h is bounded and continuous, the integrand on the left handside is equal to (Lt Ã Lt)(h), which gives the result. By a Monotone Class Theorem (e.g.Corollary 4.4 in the Appendix of Ethier and Kurtz [23]), the same holds for all boundedand measurable h.67We now turn to part (a). Let Ï : R â R be bounded and Borel. We recall from theIntroduction (see (2.1.5)) that Proposition 4.5 of [69] states thatlimÎ»ââtÎ»0EXX0(LÎ»t (Ï))= C2.1.4â«â«Ï(x0 +âtz) exp(â1tâ«F (z + tâ1/2(x0 â y0) dX0(y0))Ï0(z) dm(z) dX0(x0).(2.4.2)(The fact that the constant appearing in Proposition 4.5 of [69] equals C2.1.4 is implicit inthe proof.) The proof uses the Palm measure formula for Xt under PXX0; see Theorem 4.1.3of Dawson-Perkins [15]. The corresponding Palm measure formula for the superprocessunder N0 is in fact simpler, and the same proof shows thatlimÎ»ââN0(LÎ»t (Ï)) = C2.1.4 tâÎ»0â«Ï(âtz)Ï0(z) dm(z). (2.4.3)Consider now a bounded and continuous function Ï; we can also clearly assume that Ï â¥ 0.By Theorem 2.1.1 (under N0), Lλt (Ï) converges in L2 with respect to the probability measureN0(Xt â · |Xt > 0), which implies that it also converges in L1, allowing us to exchange limitand expectation in (2.4.3), which gives part (a) for bounded and continuous Ï. This extendsto all bounded and measurable Ï by a monotone class argument (as above for part (b)).Finally, it is clear that both (a) and (b) hold for general non-negative functions by theMonotone Convergence Theorem.We now describe how to ascertain the existence of Lt when Xt is a super-Brownianmotion under PXX0 via the cluster representation. In particular, we recall (2.1.15) and(2.1.16). Let X0 âMF (R) and t > 0.Proof of Theorem 2.1.1 for PXX0 . Let N,x1, . . . , xN ,X1t , . . . XNt be as in the cluster decom-position (2.1.16). For λ > 0, define the measure Lλt via (2.1.6) using Xt. For i = 1, . . . , N ,let Li,λt denote the measure defined in (2.1.6) corresponding to Xit . By Theorem 2.1.1 forN0 and translation invariance, Nxi(Xit â · |Xit > 0)-a.s. there exists Lit such that Li,λt â Ltin MF (R) in measure. Define Lt âMF (R) by (2.1.17). That is,dLt(x) =Nâi=11(âj 6=iXj(t, x) = 0)dLit(x).Let Ï : Râ R be bounded and continuous. We will show thatLλt (Ï)â Lt(Ï) in probability as Î»ââ. (2.4.4)68Once we establish (2.4.4), the proof of Theorem 2.1.1 for N0 applies and shows that Lλt â Ltin probability in MF (R) as Î»ââ. With the exception of L2 convergence, which we showafterward, this proves Theorem 2.1.1 for PXX0 .Turning to (2.4.4), we will argue conditionally on (N,x1, . . . , xN ). That is, we argueunder the regular conditional distribution for (X1t , . . . ,XNt ) given (N,x1, . . . , xN ). As such,we treat N â¥ 1 and x1, . . . , xN â R as fixed, and X1t , . . . ,XNt are independent randommeasures with respective laws Nxi(Xt â · |Xt > 0) for i = 1, . . . , N . Let E denote theexpectation of a probability realizing this conditional representation for Xt. ExpandingLλt (Ï) in terms of the clusters, we haveLλt (Ï) =â«Î»2Î»0X(t, x)eâÎ»X(t,x) Ï(x) dx=â«Î»2Î»0[Nâi=1Xi(t, x)]eâÎ»âNi=1Xi(t,x)Ï(x) dx=Nâi=1â«Î»2Î»0Xi(t, x)eâÎ»Xi(t,x)[eâÎ»âj 6=iXj(t,x)Ï(x)]dx=Nâi=1Li,Î»t(Ï · eâÎ»ZiN (t,·)), (2.4.5)where we define ZiN (t, x) =âj 6=iXj(t, x), in which the indices are understood to sum from1 to N . Using this notation, Lt(Ï) =âNi=1 Lit(Ï Â·1(ZiN (t, Â·) = 0)). Thus by (2.4.5), to prove(2.4.4) it is clearly enough to show that for any 1 â¤ i â¤ N ,Li,Î»t(Ï · eâÎ»ZiN (t,·))â Lit(Ï · 1(ZiN (t, Â·) = 0)) in probability as Î»ââ. (2.4.6)Without loss of generality, assume that λ > 1. Let 1 â¤ Î»â² â¤ λ. Then|Li,λt (Ï · eâÎ»ZiN (t,·))â Lit(Ï · 1(ZiN (t, Â·) = 0))|â¤ |Li,λt (Ï · (eâÎ»ZiN (t,·) â eâÎ»â²ZiN (t,·)))|+ |Li,λt (Ï · eâÎ»â²ZiN (t,·))â Lit(Ï · eâÎ»â²ZiN (t,·))|+ |Lit(Ï · (eâÎ»â²ZiN (t,·) â 1(ZiN (t, Â·) = 0)))|â¤ âÏââ|Li,Î»t (eâÎ»â²ZiN (t,·)1(ZiN (t, Â·) > 0))| + |Li,λt (Ï · eâÎ»â²ZiN (t,·))â Lit(Ï · eâÎ»â²ZiN (t,·))|+ âÏââLit(eâÎ»â²ZiN (t,·)1(ZiN (t, Â·) > 0))=:âÏââR1(Î»â², λ) +R2(Ï, Î»â²) + âÏââR3(Î»â², λ). (2.4.7)We first consider R1. Since Xit and ZiN (t, Â·) are independent and Lλt is a measurable function69of Xit , conditional on Xit we have, for all λ > 1 and 1 â¤ Î»â² < Î»,E(R1(Î»â², λ) |Xit ) =â«E(eâÎ»â²ZiN (t,x)1(ZiN (t, x) > 0)) |Xit ) dLi,λt (x)=â«E(eâÎ»â²âj 6=iXj(t,x)1(âj 6=iXj(t, x) > 0)) dLi,λt (x)â¤âj 6=iâ«Nxj(eâÎ»â²Xj(t,x)1(Xj(t, x) > 0) |Xjt > 0) dLi,λt (x)=âj 6=iNxj(Xjt > 0)â1â«Nxj(eâÎ»â²Xj(t,x) â 1(Xj(t, x) = 0)) dLi,λt (x)= (t/2)âj 6=iâ«Nxj(1â 1(Xj(t, x) = 0))â Nxj(1â eâÎ»â²Xj(t,x)) dLi,λt (x)= (t/2)âj 6=iâ«Vât (xâ xj)â V Î»â²t (xâ xj) dLi,λt (x), (2.4.8)where in the second last line we have used (2.3.11), and the last follows from (2.3.8), (2.3.10),and translation invariance. We apply (2.3.15) to the integrand and take the expectation ofthe above to obtain thatE(R1(Î»â², λ)) â¤ CN â 12t1/2âÎ»0Î»â²â(2Î»0â1)Nxi(Li,Î»t (1) |Xit > 0)= CN â 14t3/2âÎ»0Î»â²â(2Î»0â1)Nxi(Li,Î»t (1)) (by (2.3.11))â¤ C(t,N)Î»â²â(2Î»0â1), (2.4.9)for all λ > 1 and 1 â¤ Î»â² < λ, where the last inequality is by Theorem 2.1.5(a) and the factthat Li,λt (1)â Lit(1) in L2(Nxi) (from Theorem 2.1.1). Next we consider R3. Note that wecan expand and bound this term in exactly the same way as we did R1 in (2.4.7) but withLit replacing Li,λt . Taking the expectation and proceeding as above then givesE(R3) â¤ N â 14t3/2âÎ»0Nxi(Lit(1))Î»â²â(2Î»0â1). (2.4.10)Fix δ > 0. By (2.4.9) and (2.4.10) and Markovâs inequality there exists λ¯(δ) such that forÎ»â² â¥ λ¯(δ),P (R1(Î»â², λ) > δ) + P (R3(Î»â², λ) > δ) < C â²(t,N)Î»â²â(2Î»0â1)/Î´. (2.4.11)Now consider R2(Ï). Since Ï · eâÎ»â²ZiN (t,·) is a bounded, continuous function for all Î»â² â¥ 1,by Theorem 2.1.1 for Nxi , R2(Ï, Î»â²)â 0 in probability as Î»ââ for all Î»â² â¥ 1. From thisand (2.4.11) we conclude, by choosing Î»â² â¤ λ sufficiently large, that (2.4.7) converges to 0in probability as Î»ââ. As we noted in (2.4.6), this is sufficient to prove the result.70It remains to show that Lλt (Ï) â Lt(Ï) in L2(PXX0) for all continuous and boundedfunctions Ï. Let Ï be such a function, and suppose that Xt is realized as in (2.1.16) undera probability PXX0 . Under PXX0(Â· |N), from (2.4.5) and (2.1.17) we have(Lλt (Ï)â Lt(Ï))2 =( Nâi=1Li,Î»t (eâÎ»ZiN (t,·) · Ï)â Lit(1(ZiN (t, Â·) = 0) · Ï))2â¤ NNâi=1[Li,Î»t (eâÎ»ZiN (t,·) · Ï)â Lit(1(ZiN (t, Â·) = 0) · Ï)]2. (2.4.12)We recall that X1t , . . . ,XNt are iid with distribution NX¯0(Xt â · |Xt > 0), where X¯0 =X0(Â·)/X0(1) and NX0(Â·) =â«Nx(Â·)dX0(x). This implies that the N summands in (2.4.12)are identically distributed; in particular, conditional on N we define identically distributedrandom variables eN,λi â¥ 0, for i = 1, . . . , N , byeN,λi =[Li,λt (eâÎ»ZiN (t,·) · Ï)â Lit(1(ZiN (t, Â·) = 0) · Ï)]2. (2.4.13)By (2.4.6), eN,λi converges to 0 in probability as Î»ââ when conditioned on (x1, . . . , xN ).However, one can integrate the conditional probabilities over (x1, . . . , xN ) â RN to deter-mine thateN,λi â 0 in probability under PXX0(Â· |N) as Î»ââ. (2.4.14)It is clear from (2.4.13) that for all λ > 0,eN,λi â¤ 2âÏâ2â(Li,Î»t (1)2 + Lit(1)2) â i = 1, . . . , N, âN â¥ 1. (2.4.15)By Theorem 2.1.1 for N0, Li,λt (1)2 â Lit(1)2 in probability under NX¯0(Xt â · |Xt > 0)and hence under PXX0(Â· |N). Furthermore, since Li,λt (1) â Lit(1) in L2(NX¯0(Â· |Xt > 0))(by Theorem 2.1.1 for N0), it follows from Cauchy-Schwarz that Li,λt (1)2 â Lit(1)2 inL1(NX¯0(Â· |Xt > 0)); since Xit has distribution NX¯0(Xt â · |Xt > 0) under PXX0(Â· |N),this implies Li,λt (1)2 â Lit(1)2 in L1(PXX0(Â· |N)). Hence {2âÏââ(Li,Î»t (1)2 + Lit(1)2) : λ â¥ 1}is uniformly integrable. Thus by (2.4.15), {eN,λi : λ â¥ 1} is uniformly integrable, and by(2.4.14) we have L1 convergence. That is,EXX0(eN,λi |N)â 0 as Î»ââ. (2.4.16)Conditioning on N = n and summing over n â N, by (2.4.12) and Fubiniâs Theorem we71haveEXX0((Lλt (Ï)â Lt(Ï))2) â¤âân=1PXX0(N = n)nnâi=1EXX0(en,Î»i |N = n).Since EXX0(eN,λi |N) â¤ 2âÏââEXX0(Li,Î»t (1)2 + Lt(1)2) â¤ C(t, Ï) for all λ â¥ 1, for someconstant C(t, Ï) > 0 (by uniform integrability), the nth term in the above is bounded aboveby C(t, Ï)PXX0(N = n)n2. Dominated Convergence therefore allows us to exchange limitand summation in the above, which by (2.4.16) gives the result.Proof of Theorems 2.1.8 and 2.1.9. The proof of Theorem 2.1.8 is in fact implicit in theabove proof of Theorem 2.1.1 for PXX0 . Conditionally on the number of clusters N , Lt wasdefined under PXX0 by (2.1.17), so by construction it has the claimed conditional represen-tation.The proof of Theorem 2.1.9 is virtually identical to that of Theorem 2.1.8, except inthis case we already know that Lt exists and Lλt â Lt inMF (R). One can then decomposeXt in terms of the different contributions and show that Lt has the desired representationusing the same argument as appears above, making the obvious changes between the lawof super-Brownian motion and canonical measure where necessary.As we have commented on, the expression in Theorem 2.1.4, which is the same as (2.1.9)in Theorem 2.1.5(b), is finite for all bounded h, despite the appearance of non-integrability(since λ0 > 1/2). Proposition 2.1.6, which we restate here for convenience, provides a usefulupper bound on second moments which is our main tool for studying Lt. The bound is notdifficult to obtain. Its derivation relies only on trivial upper bounds and several changesof variables. Recall that EYz denotes the expectation of a standard Ornstein-Uhlenbeckprocess Y with Y0 = z.Proposition 2.1.6. For a non-negative Borel function h : R2 â R,N0((Lt Ã Lt)(h)) â¤ C22.1.4â« t0wâ2Î»0[ â«â«EYz1(exp(ââ« log(t/w)0F (Yu) du)Ã h(âtYlog(t/w),âtYlog(t/w) +âw(z2 â z1)))Ï0(z1)Ï0(z2) dm(z1) dm(z2)]dw.(2.1.10)Moreover,N0(Lt(1)2) â¤ C22.1.4Î¸21â λ0 t1â2Î»0 . (2.1.11)Proof. Let h : R2 â R be Borel measurable and non-negative. We use the formula for72N0((Lt Ã Lt)(h)) given by (2.1.9). We recall that Ï(z1, z2) â¤ 1 and use this bound, and webound above by using Vâ,âu (x, y) â¥ Vâu (x) in the exponential. This givesN0((Lt Ã Lt)(h))â¤ C22.1.4â« t0(tâ s)â2Î»0[ â«â«EB0(exp(ââ« s0Vâtâu(âtâ s z1 +Bs âBu) du))Ã h(âtâ s z1 +Bs,âtâ s z2 +Bs)Ï0(z1)Ï0(z2) dm(z1) dm(z2)]ds.Since z1 â¼ m,âtâ s z1 has a normal distribution with variance tâ s, and we interpret itas the Brownian increment Bt âBs. Hence the above is equal toC22.1.4â« t0(tâ s)â2Î»0[ â«EB0(exp(ââ« s0Vâtâu(Bt âBu) du)Ã h(Bt,âtâ s z2 +Bs)Ï0(Bt âBsâtâ s))Ï0(z2) dm(z2)]ds= C22.1.4â« t0wâ2Î»0[ â«EW0(exp(ââ« twVâu (Wu) du)h(Wt,âw z2 +Wt âWw)Ã Ï0(Wwâw)Ï0(z2) dm(z2))]dw,where in the second line we have used w = tâ s and defined Wu = Bt â Btâu. Hence Wuis a standard Brownian motion under PW0 . Recall that Vâu (x) = uâ1F (uâ1/2x). Applyingthis and letting u = er in the integral, we obtain that the above is equal toC22.1.4â« t0wâ2Î»0[ â«EW0(exp(ââ« log tlogwF (eâr/2Wer) dr)h(Wt,âw z2 +Wt âWw)Ã Ï0(Wwâw)Ï0(z2) dm(z2))]dw.We now define a stationary Ornstein-Uhlenbeck process Y (with stationary measure m) byYr = eâr/2Wer for r â R. Recall that we denote its law by EY . The above is thereforeequal toC22.1.4â« t0wâ2Î»0[ â«EY(exp(ââ« log tlogwF (Yu) du)h(ât Ylog t,âw z2 +ât Ylog t ââw Ylogw)Ã Ï0(Ylogw)Ï0(z2) dm(z2))]dw.73By stationarity of Y , we can shift time by logw in the above to obtainC22.1.4â« t0wâ2Î»0[ â«EY(exp(ââ« log(t/w)0F (Yu) du)h(ât Ylog(t/w),âw z2 +ât Ylog(t/w) ââw Y0)Ã Ï0(Y0)Ï0(z2) dm(z2))]dw.Y0 has distribution m, so we condition on the value of Y0 and call it z1. This gives thedesired expression and proves that (2.1.10) holds. The proof of (2.1.11) is a consequence ofthe following lemma.Lemma 2.4.2. For t > 0, â«P Yz (ÏF > t)Ï0(z) dm(z) = Î¸eâÎ»0t.Returning to (2.1.11), we apply (2.1.10) with h = 1. Separating the integrals, we obtainthatN0(Lt(1)2) â¤ C22.1.4 Î¸â« t0wâ2Î»0(â«P Yz(ÏF > log(t/w))Ï0(z) dm(z))dw,where we have usedâ«Ï0dm = θ. The inequality (2.1.11) now readily follows from Lemma 2.4.2,which completes the proof of Proposition 2.1.6.Proof of Lemma 2.4.2. Expanding in terms of the transition densities, we haveâ«P Yz(ÏF > t)Ï0(z) dm(z) =â« (â«qt(z, y) dm(y))Ï0(z) dm(z)= ãqt, 1 â Ï0ãL2(mÃm), (2.4.17)where ã· , Â·ãL2(mÃm) denotes the inner product on L2(m Ãm) and â is the tensor productof functions. Recall from that Theorem 2.2.1(a) that the eigenfunction expansion (2.2.1)converges in L2(mÃm) to qt(Â·, Â·), and that âÏ0âL2(m) = 1. Thus by the above and Fubiniâstheorem, (2.4.17) is equal toâân=0eâÎ»ntãÏn â Ïn, 1 â Ï0ãL2(mÃm) = eâÎ»0tãÏ0 â Ï0, 1â Ï0ãL2(mÃm)= eâÎ»0tâ«Ï20 dmâ«Ï0 dm = Î¸eâÎ»0t,where the first equality follows from orthogonality of the eigenfunctions, which implies thatâ«ÏnÏ0 dm = 0 for all n â¥ 1. The last line usesâ«Ï0 dm = θ andâ«Ï20 dm = 1.74We now use the bounds in Proposition 2.1.6 to derive the remaining properties of Ltand their consequences. In order of presentation, we now prove Theorem 2.1.3, Theorem2.1.7, and Theorem 1.2.Proof of Theorem 2.1.3. Recall that for p > 0, hp(x, y) = |xâ y|âp. We first establish thatN0((Lt Ã Lt)(hp)) <â (2.4.18)for all p < 2â 2λ0. Applying (2.1.10) with hp, we haveN0((Lt Ã Lt)(hp)) â¤C22.1.4â« t0wâ2Î»0[ â«â«EYz1(exp(ââ« log(t/w)0F (Yu) du))Ã |âw(z2 â z1)|âp Ï0(z1)Ï0(z2) dm(z1) dm(z2)]dw=C22.1.4â« t0wâ2Î»0âp/2[ â«â«EYz1(exp(ââ« log(t/w)0F (Yu) du))Ã |z1 â z1|âpÏ0(z1)Ï0(z2) dm(z1) dm(z2)]dw.Recalling (2.1.4), the expectation is equal to the survival probability P Yz1(ÏF > log(t/w)),so the above equalsC22.1.4â« t0wâ2Î»0âp/2[ â«â«P Yz1(ÏF > log(t/w)) â£â£z1 â z2â£â£âp Ï0(z1)Ï0(z2) dm(z1) dm(z2)]dw.Applying (2.2.5) and (2.2.8), both with δ = 1/8, this is bounded above byCâ« t0wâ2Î»0âp/2[ â«â« â£â£z1 â z2â£â£âp tâÎ»0 wλ0 ez21/4 ez22/8 dm(z1) dm(z2)]dw= C(p) tâÎ»0â« t0wâÎ»0âp/2dw.The second line follows because the integrand has Gaussian tails in z1 and z2 and p < 2â2Î»0 < 1. Finally, the integral in the final line is finite becauseâÎ»0âp/2 > âÎ»0âÎ»0+1 > â1,which proves (2.4.18). In fact, we have shown thatN0((Lt Ã Lt)(hp)) â¤ C(p)t1â2Î»0âp/2. (2.4.19)Next, we establish the same under PXX0 . That is, we will show thatEXX0((Lt Ã Lt)(hp)) <â (2.4.20)75for p < 2â 2λ0. We use the cluster decomposition and argue conditionally as in the proofof Theorem 2.1.1 (for PXX0) above. Suppose that PXX0is a probability under which Xt isrealized as in (2.1.16). Conditioning on N,x1, . . . , xN , by (2.1.17) we havedLt(x) â¤Nâi=1dLit(x).Thus we obtain thatâ«â«|xâ y|âpdLt(x) dLt(y)â¤â«â«|xâ y|âp( Nâi=1dLit(x))( Nâi=jdLjt (y))=Nâi=1â«â«|xâ y|âpdLit(x) dLit(y) +Nâi=1âj 6=iâ«â«|xâ y|âpdLit(x) dLjt (y). (2.4.21)Recall that the Xit are independent with distributions Nxi(Xt â · |Xt > 0). By (2.3.11) and(2.4.19), we therefore haveNxi(â«â«|xâ y|âpdLit(x) dLit(y)â£â£â£â£Xit > 0)= C(p)t1â2Î»0âp/2 (2/t)â1 =: C1(p) t2â2Î»0âp/2,(2.4.22)which provides a bound for the summands in the first term of (2.4.21). We now considerthe mixed integrals in (2.4.21), that is, the summands in the second term. Without loss ofgenerality, let i = 1 and j = 2, and denote their (independent) distributions by N1x1(X1t â· |X1t > 0),N2x2(X2t â · |X2t > 0). Because the integrands are non-negative, we can changethe order of integration and obtainN1x1 â N2x2(â«â«|xâ y|âpdL1t (x) dL2t (y)â£â£â£â£X1t > 0,X2t > 0)= N1x1(â«N2x2(â«|xâ y|âpdL2t (y)â£â£â£â£X2t > 0)dL1t (x)â£â£â£â£X1t > 0)(2.4.23)76To compute the inner expectation we apply translation invariance and (2.3.11), which givesN2x2(â«|y â x|âpdL2t (y)â£â£â£â£X2t > 0)= (t/2)N0(â«|y â x|âpdLt(y â x2))= (t/2)N0(â«|y â x+ x2|âpdLt(y))= C2.1.4(t/2)tâÎ»0â«|âtz â (xâ x2)|âpÏ0(z) dm(z),where the last line follows from the mean measure formula (2.1.8). By (2.2.5) with δ = 1/4,we have that Ï0(z2) dm(z2) â¤ c eâz22/4dz2. Thus the above is bounded above byCt1âÎ»0â«(|âtz â (xâ x2)|âp â¨ 1)eâz2/4 dz= Ct1âÎ»0â«(|w â (xâ x2)|âp â¨ 1)tâ1/2eâw2/4t dwâ¤ C t1âÎ»0tâ1/2â«|w â (xâ x2)|âp 1|wâ(xâx2)|â¤1 dw + C t1âÎ»0â«tâ1/2eâw2/4t dw= C â²(p)t1/2âÎ»0 + Ct1âÎ»0 <â.By the above bound and another application of (2.1.8), (2.4.23) is bounded above by[C â²(p)t1/2âÎ»0 + Ct1âÎ»0]N1x1(L1t (1) |X1t > 0) =: C2(p)[t3/2â2Î»0 + t2â2Î»0]. (2.4.24)We note that both (2.4.22) and (2.4.24) are independent of the points x1, . . . , xN . Thereforeby these bounds and (2.4.21) we have shown thatEXX0((Lt Ã Lt)(hp) |N) â¤ C1(p)Nt2â2Î»0âp/2 + C2(p)(N2 âN)[t3/2â2Î»0 + t2â2Î»0].Taking the expectation above with respect to N , which we recall is Poisson with mean2X0(1)/t, givesEXX0((Lt Ã Lt)(hp)) â¤ C1(p)X0(1) t1â2Î»0âp/2 + C2(p)X0(1)2[tâ1/2â2Î»0 + tâ2Î»0]<â,(2.4.25)which proves (2.4.20).Under both PXX0 and N0, we have shown that the p-energy of Lt has finite expectation,and hence Lt has finite p-energy almost surely, for all p < 2â2Î»0. By the energy method (see,for example, Theorem 4.27 of Mo¨rters and Peres [65]), this implies that dim(BZt) â¥ 2â2Î»0a.s. on {Lt > 0} under PXX0 and N0. Combined with Theorem 2.A, this completes the proof77of Theorem 2.1.3 for PXX0 . To see that the upper bound on the dimension holds for N0 followsfrom the cluster decomposition. Consider Xt under PXδ0. In the cluster decomposition ofXt, the probability that N = 1 is positive. Conditioning on this event, Xt is equal to X1t ,which has law N0(X1t â · |Xt > 0). Because dim(BZt) â¤ 2 â 2λ0 a.s. on this event, wetherefore have N0({dim(BZt) â¤ 2 â 2λ0} â£â£Xt > 0) = 1. In particular, this implies thatN0({dim(BZt) â¤ 2 â 2λ0} â£â£Lt > 0) = 1, since {Lt > 0} â {Xt > 0}. We note thatconditioning on the event {Lt > 0} is valid under both PXX0 and N0 by Theorem 2.1.2(a).This completes the proof.Proof of Theorem 2.1.7. To see part (a), we note that (2.4.2) gives an expression forlimÎ»ââEXX0(LÎ»t (Ï)). On the subsequence {Î»n}ân=1 from Theorem 2.1.1, Lλnt (Ï) â Lt(Ï)a.s. for bounded and continuous Ï, so it is enough to show that limnââEXX0(LÎ»nt (Ï)) =EXX0(limnââLÎ»nt (Ï)). By Theorem 2.1.1, Lλt (Ï) converges in L2(PXX0) and hence is boundedin L2(PXX0). It is therefore uniformly integrable, which justifies the above exchange of limitand integration. This proves the result for bounded and continuous Ï. We extend themoment formula to bounded measurable functions by a Monotone Class Lemma and tonon-negative measurable functions by Monotone convergence.We now prove part (b). Suppose we realize Xt under a probability PXX0such that (2.1.16)holds. Conditionally on N , by (2.1.17) we haveLt(1)2 â¤( Nâi=1Lit(1))2=Nâi=1Lit(1)2 +Nâi=1âj 6=iLit(1)Ljt (1).The clusters are independent with laws NX¯0(Xit â · |Xit > 0) = (t/2)NX¯0({Xit > 0,Xit â Â·}),the equality by (2.3.11). Thus, applying Theorem 2.1.5(a) and Proposition 2.1.6(b) to theabove and using independence, we obtainEXX0(Lt(1)2 |N) â¤ CN(t/2)t1â2Î»0 +C(N2 âN)(t/2)2tâ2Î»0 . (2.4.26)As in the proof of Theorem 2.1.3, we take the expectation with respect to N , which has aPoisson(2X0(1)/t) distribution. This proves part (b).It remains to prove Theorem 2.1.2. We will derive part (a) below using Proposition2.1.6; part (b) requires a few lemmas which we now discuss.We say that Lt has an atom of mass c > 0 at x if Lt({x}) = c. We decompose Lt asLt = LËt + νt, (2.4.27)where LËt is atomless and νt is strictly atomic. We begin with an elementary observation78which provides an upper bound for the mass of the atoms of a measure. Let M â N.Let In1 = [âM,âM + 2ân], and for k = 2, 3 . . . , 2M2n, define the dyadic interval Ink =(âM + (k â 1)2ân,âM + k2ân]. Then {Ink : k â¤ 2M2n} is a partition of [âM,M ] intodisjoint intervals of length 2ân. The following lemma is elementary.Lemma 2.4.3. Fix M â N and suppose that µ is a finite measure supported on [âM,M ]with decomposition µ = Ï + ν, where Ï is atomless and ν =âiâI ciδxi is strictly atomic.Then for every n â¥ 1,2M2nâk=1µ(Ikn)2 â¥âiâIc2i .The next lemma gives an upper bound for the second moment of Lt on a ball. Wedenote by B(x, r) the ball of radius r > 0 centred at x â R. We recall sâ(Î´) from Theorem2.2.1(c); in what follows we use δ = 1/8, and sâ denotes sâ(1/8).Lemma 2.4.4 (Second moments on balls). There is a constant C2.4.4 > 0 and t-dependentconstant C2.4.4(t) > 0 such that for all x â R and r < eâsât,N0(Lt(B(x, r))2) â¤ C2.4.4[tâÎ»0r2â2Î»0 PW0 (W4t/3 â B(x, r)) + tâ3Î»0+1/2r PW0 (Wt â B(x, r))]â¤ C2.4.4(t)[r3â2Î»0 + r2],where W is a standard Brownian motion under PW0 .We delay the proof of this lemma to the end of the section and first prove Theorem2.1.2.Proof of Theorem 2.1.2. First consider part (a). For canonical measure, via the secondmoment method we haveN0 (Lt(1) > 0) â¥ (N0(Lt(1)))2N0(Lt(1)2)â¥ C22.1.4Î¸2tâ2Î»0C22.1.4Î¸2t1â2Î»0(1â Î»0)â1 =1â λ0t,where we recall thatâ«Ï0 dm = θ and we have used Theorem 2.1.5(a) and (2.1.11). Werecall that N0 (Xt > 0) = 2/t, which implies that N0(Lt > 0â£â£Xt > 0) â¥ 1âÎ»02 . This provesthe result for N0.To see that PXX0(Lt > 0) > 0, we realize Xt under PXX0via a cluster decomposition. Theevent that the number of clusters N is exactly one has some positive probability p > 0;restricted to this event, Xt is equal to a single canonical cluster conditioned on survival (asin the proof of Theorem 2.1.3), which we just showed has probability at least 1âÎ»02 thatLt > 0. Hence PXX0(Lt > 0) â¥ p1âÎ»02 > 0.79We now prove part (b). First consider Lt under N0 and recall the decomposition (2.4.27),i.e. Lt = LËt + νt, the latter strictly atomic. Fix M â N and consider the restriction of Ltto [âM,M ], i.e. dL(M)t (x) := 1[âM,M ](x) dLt(x), with decomposition L(M)t = LË(M)t + ν(M)t .Note that the radius of the dyadic intervals is r(Ikn) = r = 2â(n+1). By Lemma 2.4.4, wehaveN0(2M2nâk=1L(M)t (Ikn)2)=2M2nâk=1N0(L(M)t (Ikn)2)â¤ C(t) 2M2n[(2â(n+1))3â2Î»0 + (2â(n+1))2]â¤ C(t) 2M[(2ân)2â2Î»0 + 2ân]â 0 as nââbecause 2 â 2λ0 > 0. Moreover, by Lemma 2.4.3, the first expression is greater than orequal to the expectation (under N0) of the sum of the squares of the atoms of L(M)t . Theabove implies that this expectation must in fact be zero, so ν(M)t = 0 N0-a.s. As this holdsfor all M , νt = 0 and Lt is atomless under N0. To obtain the result under PXX0, we notefrom the cluster decomposition and (2.1.17) that (conditionally) Lt is a sum of N measureswhich are atomless by the above, and hence is atomless.Proof of Lemma 2.4.4. We apply (2.1.10) with h(z1, z2) = 1B(x,r)(z1) 1B(x,r)(z2). This givesN0(Lt(B(x, r))2)= Câ« t0wâ2Î»0[ â«â«EYz1(exp(ââ« log(t/w)0F (Yu) du)Ã 1B(x,r)(âtYlog(t/w))1B(x,r)(âtYlog(t/w) +âw(z2 â z1)))Ï0(z1)Ï0(z2)dm(z1)dm(z2)]dw.(2.4.28)We now divide the above into two cases depending on the size of w. We first consider thesingular case, where w is small.Case 1: w < eâsât.We interpret the exponential in (2.4.28) as the probability that Y survives until timelog(t/w) when it is subject to Markovian killing with rate F (Yu). Because this probabilityis equal to the integral of the transition density over all of R, the portion of the integral80corresponding to w â [0, eâsât] equalsCâ« eâsâ t0wâ2Î»0[ â«â«â«qlog(t/w)(z1, y) 1B(x,r)(âty) 1B(x,r)(âty +âw(z2 â z1))Ã Ï0(z1)Ï0(z2) dm(z1) dm(z2) dm(y)]dw.â¤ Câ« eâsâ t0wâ2Î»0[ â«â«â«eâÎ»0 log(t/w) ez21/8ey2/8 1B(x,r)(âty) 1B(x,r)(âty +âw(z2 â z1))Ã Ï0(z1)Ï0(z2) dm(z1)dm(z2) dm(y)]dw.â¤ CtâÎ»0â« eâsâ t0wâÎ»0â«ey2/81B(x,r)(âty)Ã[ â«â«ez21/4ez22/8 1B(x,r)(âty +âw(z2 â z1)) dm(z1) dm(z2)]dm(y) dw.(2.4.29)The first inequality uses (2.2.2) with δ = 1/8, which applies because log(t/w) > sâ for allw in the above integral, and the second uses (2.2.5), both with δ = 1/8. In the integral inthe last line we collect all the Gaussian terms. The square-bracketed term is equal toCâ«â«1B(x,r)(âty +âw(z2 â z1)) eâz21/4eâ3z22/8 dz1 dz2= C â²â«1B(x,r)(âty +âwz) eâ3z2/20 dz.We have used the convolution property for independent Gaussians. We define Gaussianrandom variables g1 â¼ N (0, 4t/3) and g2 â¼ N (0, 10/3). Substituting the last expressioninto (2.4.29), we obtainCtâÎ»0â« eâsâ t0wâÎ»0[ â«â«1B(x,r)(âty) 1B(x,r)(âty +âwz) eâ3z2/20 eâ3y2/8dz dy]dw= C â²tâÎ»0â« eâsâ t0wâÎ»0[P(g1 â B(x, r), g1 +âwg2 â B(x, r))]dwâ¤ CtâÎ»0â« eâsâ t0wâÎ»0[P(g1 â B(x, r))P(âwg2 â B(0, 2r))]dw= CtâÎ»0P(g1 â B(x, r)) â« eâsâ t0wâÎ»0P(g2 â B(0, 2rwâ1/2))dw. (2.4.30)Suppose that 4r2 < eâsât. If 2rwâ1/2 > 1, we bound the probability in the integral above by1. If 2rwâ1/2 â¤ 1, the probability is simply bounded by the diameter of the ball, 4rwâ1/2.81Thus (2.4.30), and hence (2.4.29), is bounded above byCtâÎ»0P (g1 â B(x, r))[ â« 4r20wâÎ»0dw + 4râ« eâsâ t4r2wâÎ»0â1/2dw]= CtâÎ»0P (g1 â B(x, r))[41âÎ»0r2â2Î»01â λ0 +4rλ0 â 1/2((4r2)â(Î»0â1/2) â (teâsâ)â(Î»0â1/2))]â¤ CtâÎ»0P (g1 â B(x, r)) r2â2Î»0 . (2.4.31)Finally, note that if 4r2 â¥ eâsât, then (2.4.30) is bounded above byCtâÎ»0P (g1 â B(x, r))â« eâsâ t0wâÎ»0dw â¤ CtâÎ»0P (g1 â B(x, r))(eâsât)1âÎ»0â¤ CtâÎ»0P (g1 â B(x, r))r2â2Î»0 ,so the upper bound for (2.4.29) obtained in (2.4.31) holds in this case as well.Case 2: w â (eâsât, t].In this case we simply bound the exponential term in (2.4.28) above by 1, effectivelyignoring the killing, in which case Ylog(t/w) â¼ m. We also use (2.2.5) with δ = 14 . Hence thecontribution to (2.4.28) from the w â (eâsât, t] case is bounded above byCâ« teâsâ twâ2Î»0[ â«â«â«1B(x,r)(âty) 1B(x,r)(âty +âw(z2 â z1))Ã ez21/4 ez22/4 dm(z1) dm(z2) dm(y)]dwâ¤ Câ« teâsâ twâ2Î»0[ â«â«â«1B(x,r)(âty) 1B(x,r)(âty +âw(z2 â z1))Ã eâz21/4 eâz22/4 dz1 dz2 dm(y)]dw= Câ« teâsâ twâ2Î»0[ â«â«1B(x,r)(âty) 1B(x,r)(âty +âwz) eâz2/2 dz dm(y)]dwâ¤ CtâÎ»0P (g3 â B(x, r))â« teâsâ twâ2Î»0P (g4 â B(0, 2rwâ1/2)) dw.In the above, g3 â¼ N (0, t) and g4 â¼ N (0, 1). The third line follows by the convolutionproperty of Gaussians. We again bound the probability in the integral by the size diameter82of the ball, which gives the following upper bound for the above:CtâÎ»0P (g3 â B(x, r)) 4râ« teâsâ twâ2Î»0â1/2 dw.â¤ CtâÎ»0P (g3 â B(x, r)) 4r (eâsât)â2Î»0+1/2= Ctâ3Î»0+1/2P (g3 â B(x, r)) r. (2.4.32)By combining (2.4.31) and (2.4.32) and interpreting the Gaussian probabilities in terms ofBrownian motion, we obtain the first inequality of the result. The second bound is obtainedby bounding the Brownian density above by its maximum value.2.5 Proof of Theorem 2.1.4The proof of Theorem 2.1.4 is split up into two main parts. In the first, we obtain represen-tations for N0((Lλt ÃLÎ»â²t )(h)) in terms of solutions to (2.3.4), in particular the family VÎ»,Î»â²introduced in Section 2.3. In the second part, using these representations, we establish con-vergence of N0((Lλt ÃLÎ»â²t )(h)) as λ, Î»â² ââ. The proof of a technical lemma (Lemma 2.5.5)is given in Section 2.6.2.5.1 PDE representations and preliminary boundsWe begin by deriving an expression for second moments of Lλt under the canonical measure.In particular, we study N0((LÎ»tÃ, LÎ»â²t )(h)) for λ, Î»â² > 0. The formula we obtain is naturallysuggested by a branching particle heuristic. Its proof uses PDE methods. Let EBx denote theexpectation of a Brownian motion started at x. EB1,B2(x,y) denotes the law of two independentBrownian motions B1 and B2 started from points x and y respectively. We recall thedefinition of V Î»,Î»â²t from (2.3.20).Proposition 2.5.1. Let h : R2 â R be a bounded Borel function and λ, Î»â², t > 0. ThenN0((Lλt Ã LÎ»â²t )(h)) = (Î»Î»â²)2Î»0â« t0EB0(EB1, B2(0,0)[h(Bs +B1tâs, Bs +B2tâs)Ã exp(ââ« s0V Î»,Î»â²tâu (B1tâs +Bs âBu, B2tâs +Bs âBu) du)Ã exp(ââ« tâs0V Î»,Î»â²r (B1r , B1r +B2tâs âB1tâs) dr)Ã exp(ââ« tâs0V Î»,Î»â²r (B2r +B1tâs âB2tâs, B2r ) dr)])ds.The proof of Proposition 2.5.1 requires the following lemma.83Lemma 2.5.2. Let Ï â MF (R) and Ï1, Ï2 â L1(R) be non-negative and continuous. ThenN0(Xt(Ï1)Xt(Ï2)eâXt(Ï))=â« t0EB0(exp(ââ« s0V Ïtâu(Bu) ds)Ãâi=1,2EBi0[exp(ââ« tâs0V Ïtâsâr(Bs +Bir) dr)Ïi(Bs +Bitâs)])ds.Proof. Let Ç«1, Ç«2 > 0 and Ï,Ï1 and Ï2 be as in the statement. Viewing Ï1 and Ï2 as thedensity functions of the finite measures they induce (i.e. Ïi(A) =â«A Ïi(x) dx), let VÏ,Ç«1,Ç«2tdenote the solution to (2.3.4) when Ï = Ï + Ç«1Ï1 + Ç«2Ï2 â MF (R). By (2.3.3) and thediscussion below (2.3.4),N0(1â eâXt(Ï+Ç«1Ï1+Ç«2Ï2)) = V Ï,Ç«1,Ç«2t (0).We differentiate this expression once with respect to Ç«1 and once with respect to Ç«2. Thederivatives of the inner expression of the left hand side are bounded above by integrablequantities (i.e. Xt(Ï1) and Xt(Ï1)Xt(Ï2)) so we can take the differentiation inside theexpectation in the probabilistic representation, and the derivatives of the right hand sideexist. The resulting equation is the following:N0(Xt(Ï1)Xt(Ï2) eâXt(Ï+Ç«1Ï1+Ç«2Ï2))= â â2âÇ«1âÇ«2V Ï,Ç«1,Ç«2t (0). (2.5.1)We note that the limit of the left hand side as Ç«1, Ç«2 â 0 is the desired expression. Wenow obtain an expression for the first derivatives of V Ï,Ç«1,Ç«2t (0) with respect to Ç«1 and Ç«2.Consider the following partial differential equation:âutât=â2ut â V Ï,Ç«1,Ç«2t ut for (t, x) â (0,â)Ã R, ut â Ï1 as t â 0, (2.5.2)where the ut â Ï1 in the sense of weak convergence of measures. The above can be obtainedheuristically by formally differentiating (2.3.4) with respect to Ç«1 when the initial conditionsare Ï + Ç«1Ï1 + Ç«2Ï2. By Lemmas 2.3 and 2.5 of [70], (2.5.2) has a unique solution, whichwe denote by U1,Ç«1,Ç«2t , which satisfiesV Ï,Ç«1,Ç«2t (x) = VÏ,0,Ç«2t (x) +â« Ç«10U1,Ç«,Ç«2t (x) dǫ.Thus U1,Ç«1,Ç«2t =ââÇ«1V Ï,Ç«1,Ç«2t . We can apply the same argument to obtain a similar rep-resentation for ââÇ«2VÏ,Ç«1,Ç«2t , which we denote by U2,Ç«1,Ç«2t . Both U1,Ç«1,Ç«2t and U2,Ç«1,Ç«2t haveFeynman-Kac representations; for example, see Theorem 7.6 of Karatzas and Shreve [42]84(on p. 366). For i = 1, 2 we haveU i,Ç«1,Ç«2t (x) = EBx(Ïi(Bt) exp(ââ« t0V Ï,Ç«1,Ç«2tâs (Bs) ds)). (2.5.3)We take the expression for i = 1 and differentiate it with respect to Ç«2. We obtainâ â2âÇ«2âÇ«1V Ï,Ç«1,Ç«2t (x)= EBx(Ï1(Bt) exp(ââ« t0V Ï,Ç«1,Ç«2tâs (Bs) ds)â« t0U2,Ç«1,Ç«2tâs (Bs) ds)= EBx(Ï1(Bt) exp(ââ« t0V Ï,Ç«1,Ç«2tâs (Bs) ds)Ãâ« t0EB20(Ï2(Bs +B2tâs) exp(ââ« tâs0V Ï,Ç«1,Ç«2tâsâr (Bs +B2r ) dr)ds)),where the final line follows from another application of (2.5.3), this time with i = 2. Firstwe note that all the terms are non-negative, so we can take the internal integral over timeoutside the expectation. For s < t, the integrand then describes one Brownian motionstarted at 0 and run to time t, and a second which branches from the first at time s andevolves independently. By applying the Markov property at time s we equivalently view itas a Brownian path that branches at time s into two independent Brownian motions B1and B2 which themselves run for a duration of tâ s. This formulation combined with theindependence of the Brownian motions gives usâ2âÇ«2âÇ«1V Ï,Ç«1,Ç«2t (x) = ââ« t0EBx(exp(ââ« s0V Ï,Ç«1,Ç«2tâu (Bu) ds)Ãâi=1,2EBi0[Ïi(Bs +Bitâs) exp(ââ« tâs0V Ï,Ç«1,Ç«2tâsâr (Bs +Bir) dr)]ds).The derivatives in Ç«1 and Ç«2 are one-sided at 0 so we cannot exactly evaluate at Ç«1 = Ç«2 = 0.However, V Ï,Ç«1,Ç«2t (x) is continuous in Ç«1 and Ç«2 and the integrand is bounded above byâÏ1âââÏ2ââ so we can apply bounded convergence. As Ç«1, Ç«2 â 0, V Ï,Ç«1,Ç«2t â V Ït byLemma 2.1(d) of [70]. We also take Ç«1, Ç«2 â 0 in the left hand side of (2.5.1) and applyDominated Convergence. Evaluating at x = 0 gives the result.Proof of Proposition 2.5.1. We will prove the result for functions of product form, i.e.h(x, y) = Ï1(x)Ï2(y), and then use a monotone class theorem. Let x1, x2 â R and λ, Î»â² > 0.Consider the expression from Lemma 2.5.2 with Ï = λδx1 +Î»â²Î´x2 . For now we simply let Ï1and Ï2 be functions satisfying the assumptions of Lemma 2.5.2, but we will shortly choose85them to be approximate identities at x1 and x2. Applying Lemma 2.5.2, we haveN0(Xt(Ï1)Xt(Ï2)eâÎ»X(t,x1)âÎ»â²X(t,x2))=â« t0EB0(exp(ââ« s0V Î»,Î»â²tâu (Bu â x1, Bu â x2) du)Ãâi=1,2EBi0[exp(ââ« tâs0V Î»,Î»â²tâsâr(Bs +Bir â x1, Bs +Bir â x2) dr)Ïi(Bs +Bitâs)])ds,(2.5.4)where we have also used (2.3.21), i.e. translation invariance of V Î»,Î»â²(x1, x2). Now letÏi = pδ(· â xi), where we recall that pδ(·) denotes the Gaussian density of variance δ. LetÏ1, Ï2 be bounded, continuous functions and integrate Ï1(x1)Ï1(x2) multiplied by the aboveover x1 and x2. The left hand side is thenâ«â«Ï1(x1)Ï2(x2)N0(Xt(pÎ´(Â· â x1))Xt(pδ(· â x2))eâÎ»X(t,x1)âÎ»â²X(t,x2))dx1 dx2. (2.5.5)The absolute value of (2.5.5) is bounded above byâÏ1âââÏ2ââN0(â«Xt(pÎ´(Â· â x1))dx1â«Xt(pÎ´(Â· â x2)) dx2), (2.5.6)where the change of order of integration follows because all the terms are non-negative oncewe bound |Ïi(xi)| by âÏiââ. Now we note thatâ«Xt(pÎ´(Â· â xi))dxi =â«â«X(t, y)pδ(xi â y) dy dxi =â«X(t, y)(â«pÎ´(xi â y) dxi)dy= Xt(1).Combined with (2.5.6), this implies that the expression in (2.5.5) is integrable and itsabsolute value is bounded above by âÏ1âââÏ2ââN0(Xt(1)2). We note that N0(Xt(1)2) <â.To see this, first observe that EXδ0(Xt(1)2) is finite and in fact equals 1+t, which follows fromthe martingale problem for super-Brownian motion (see Section II.5 of [83]). By the clusterdecomposition (2.1.16), EXδ0(Xt(1)2) is equal to the mean of a Poisson random variablemultiplied by N0(Xt(1)2|Xt > 0), and so the latter quantity, and hence N0(Xt(1)2), are alsofinite. Thus we can apply Fubini and rewrite (2.5.5) asN0(â«â«Ï1(x1)Ï2(x2)Xt(pÎ´(Â· â x1))Xt(pδ(· â x2))eâÎ»X(t,x1)âÎ»â²X(t,x2)dx1 dx2). (2.5.7)As noted, the absolute value of the expression inside N0 is bounded above byâÏ1âââÏ2ââXt(1)2, which is integrable under N0, for all δ. We take δ â 0 and apply86Dominated Convergence to obtain that the limit of (2.5.7) as δ â 0 is equal toN0(limÎ´â0+â«â«Ï1(x1)Ï2(x2)Xt(pÎ´(Â· â x1))Xt(pδ(· â x2))eâÎ»X(t,x1)âÎ»â²X(t,x2) dx1 dx2)= N0(limÎ´â0+(â«Ï1(x1)Xt(pÎ´(Â· â x1))eâÎ»X(t,x1)dx1)Ã(â«Ï2(x2)Xt(pÎ´(Â· â x2))eâÎ»â²X(t,x2)dx2)). (2.5.8)We know thatXt(pδ(· â xi)) =â«X(t, y)pδ(y â xi) dy = Xt â pδ(xi).Moreover, X(t, Â·) â Cc(R) (i.e. X(t, Â·) is continuous with compact support) N0-a.s. and{pδ}δ>0 are an approximate identity family, which together with the above imply thatXt(pÎ´(Â·âxi))â X(t, xi) as δ â 0. We note that for all δ > 0, |Xt(pÎ´(Â·âxi)| = |XtâpÎ´(xi)| â¤âX(t, Â·)ââ. Choose K â N such that supp(Xt) â [âK,K]. On the set [â2K, 2K] webound |Xt(pδ(· â xi)| above by âX(t, Â·)ââ. For |xi| > 2K, a short calculation shows that|Xt(pδ(· â xi))| â¤ C(K)Xt(1) p1(|xi| âK) for some constant C(K) > 0 for all 0 < δ â¤ 1/2.Hence |Xt(pδ(· â xi))| has an upper bound which is a bounded function with Gaussiantails, uniformly for 0 < δ â¤ 1/2. Using this bound and boundedness of Ïi, we can applyDominated Convergence in (2.5.8), which gives that the limit of (2.5.5) as δ â 0 equalsN0((â«Ï1(x1)X(t, x) eâÎ»X(t,x1)dx1)(â«Ï2(x2)X(t, x2)eâÎ»â²X(t,x2)dx2)).When rescaled by (Î»Î»â²)2Î»0 this is equal to N0(Lλt (Ï1)LÎ»â²t (Ï2)). We now turn our attentionto the right hand side of (2.5.4). With Ïi = pÎ´(Â·âxi), integrating against Ï(x1)Ï(x2)dx1dx2,we haveâ«â«Ï1(x1)Ï2(x2)(â« t0EB0(EB1,B2(0,0)[exp(ââ« s0V Î»,Î»â²tâu (Bu â x1, Bu â x2)du)Ã exp(ââ« tâs0V Î»,Î»â²tâsâr(Bs +B1r â x1, Bs +B1r â x2)dr)pδ(Bs +B1tâs â x1)Ã exp(ââ« tâs0V Î»,Î»â²tâsâr(Bs +B2r â x1, Bs +B2r â x2)dr)pδ(Bs +B2tâs â x2)])ds)dx1dx2.Since the above is equal to (2.5.5), which we have shown is integrable, we can take the spatialintegrals inside the expectations. At this point we note that we are integrating a boundedfunction of x1 and x2 with respect to the densities pδ(Bs + B1tâs â xi), which, because pδis the kernel of the Brownian semigroup, is the same as viewing xi as Bs +Bitâs+Î´. Hence87the above is equal toâ« t0EB0(EB1,B2(0,0)[Ï1(Bs +B1tâs+Î´)Ï2(Bs +B2tâs+Î´)Ã exp(ââ« s0V Î»,Î»â²tâu (Bu âBs âB1tâs+Î´, Bu âBs âB2tâs+Î´) du)Ã exp(ââ« tâs0V Î»,Î»â²tâsâr(B1r âB1tâs+Î´, B1r âB2tâs+Î´) dr)Ã exp(ââ« tâs0V Î»,Î»â²tâsâr(B2r âB1tâs+Î´, B2r âB2tâs+Î´) dr)])ds. (2.5.9)Taking δ â 0 and applying Dominated Convergence, we note that because Bitâs+Î´ â Bitâsand Ï1, Ï2 and VÎ»,Î»â²s are continuous, the limit is equal to the above with δ = 0. To obtain thedesired form we make a time reversal of the Brownian motions. Define BËiu = BitâsâBitâsâu.We note that the BËi are standard Brownian motions and that BËitâs = Bitâs, BËi0 = 0 andBir âBitâs = âBËitâsâr. Making this substitution shows that (2.5.9) with δ = 0 is equal toâ« t0EB0(EBË1,BË2(0,0)[Ï1(Bs + BË1tâs)Ï2(Bs + BË2tâs)Ã exp(ââ« s0V Î»,Î»â²tâu (Bu âBs â BË1tâs, Bu âBs â BË2tâs) du)Ã exp(ââ« tâs0V Î»,Î»â²tâsâr(âBË1tâsâr,âBË1tâsâr + BË1tâs âB2tâs) dr)Ã exp(ââ« tâs0V Î»,Î»â²tâsâr(âBË2tâsâr + BË2tâs âB1tâs,âBË2tâsâr) dr)])ds.The time index of the Brownian motions now matches the time index of the function V Î»,Î»â²in the last two lines, allowing us to reverse the time of the integrals for a simpler expression.To obtain the desired expression we now use (2.3.22), i.e. V Î»,Î»â²t (a, b) = VÎ»,Î»â²t (âa,âb), andrelabel BËi to be simply Bi. This proves the result for h(x1, x2) = Ï1(x1)Ï2(x2) when Ï1, Ï2are bounded and continuous. The result for general bounded measurable h : R2 â R nowfollows from a standard monotone class argument such as Corollary 4.4 in the Appendix ofEthier and Kurtz [23].Definition. Let ÎÎ»,Î»â²(s) denote the integrand in Proposition 2.5.1, so that the propositionstates thatN0((Lλt Ã LÎ»â²t )(h)) = (Î»Î»â²)2Î»0â« t0ÎÎ»,Î»â²(s)ds. (2.5.10)ÎÎ»,Î»â²(s) also depends on h, but we omit this. The next lemma changes variables to obtain88an expression involving Ornstein-Uhlenbeck processes. We first introduce some notation.For bounded and measurable h : R2 â R and a (continuous) path (Bu : u â [0, s]), defineÎ¨Î»,Î»â²B,s (Â·, Â·) byÎ¨Î»,Î»â²B,s (x, y) = h(x+Bs, y +Bs) exp(ââ« s0V Î»,Î»â²tâu (x+Bs âBu, y +Bs âBu) du). (2.5.11)We define Hcu as a scaling of VÎ»,Î»â²t :Hcu(x, y) = uV1,cu (âux,âuy) = Vâu,âuc1 (x, y). (2.5.12)The scaling in the following lemma cannot be done uniformly for all s â [0, t] because itrequires λ2 > (tâ s)â1 and Î»â²2 > (tâ s)â1. We derive an expression for ÎÎ»,Î»â²(s) in terms oftwo independent Ornstein-Uhlenbeck processes which we denote Y 1 and Y 2, for which wedenote the joint (independent) expectation EY1,Y 2(x,y) .Lemma 2.5.3. Let 0 < s < t, T1 = T1(s) = log(λ2(t â s)), T2 = T2(s) = log(Î»â²2(t â s)).Then for all λ > (tâ s)â1/2 and Î»â² > (tâ s)â1/2, we haveÎÎ»,Î»â²(s) = EB0(EB1, B2(0,0)[EY1,Y 2(B11 ,B21)(Î¨Î»,Î»â²B,s (âtâ s Y 1T1 ,âtâ s Y 2T2)Ã exp(ââ« 10V 1,Î»â²/Î»u (B1u, B1u + eT1/2(Y 2T2 â Y 1T1)) + V 1,Î»/Î»â²u (B2u, B2u + eT2/2(Y 1T1 â Y 2T2))du)Ã exp(ââ« T10HÎ»â²/Î»eu (Y1u , Y1u + e(T1âu)/2(Y 2T2 â Y 1T1))du)Ã exp(ââ« T20HÎ»/Î»â²eu (Y2u , Y2u + e(T2âu)/2(Y 1T1 â Y 2T2))du))]).Proof. We begin with the expression from Proposition 2.5.1. We observe that Î¨Î»,Î»â²B,s appearsand we may write the quantities in the first two lines as Î¨Î»,Î»â²B,s (B1tâs, B2tâs). In the third andfourth lines we apply (2.3.23) to obtainÎÎ»,Î»â²(s) = EB0(EB1, B2(0,0)[Î¨Î»,Î»â²B,s (B1tâs, B2tâs)Ã exp(ââ« tâs0Î»2V1,Î»â²/Î»Î»2r(Î»B1r , λ(B1r +B2tâs âB1tâs)) dr)Ã exp(ââ« tâs0Î»â²2V Î»/Î»â²,1Î»â²2r(Î»â²(B2r +B1tâs âB2tâs), Î»â²B2r ) dr)]).We define BË1u = Î»B1Î»â2u and BË2u = Î»â²B2Î»â²â2u, which are both standard Brownian motions.89Making a time change in the integrals (i.e. letting u = λ2r or Î»â²2r) givesÎÎ»,Î»â²(s) = EB0(EBË1, BË2(0,0)[Î¨Î»,Î»â²B,s (Î»â1BË1Î»2(tâs), Î»â²â1BË2Î»â²2(tâs))Ã exp(ââ« Î»2(tâs)0V 1,Î»â²/Î»u (BË1u, BË1u +Î»Î»â²BË2Î»â²2(tâs) â BË1Î»2(tâs)) du)Ã exp(ââ« Î»â²2(tâs)0V Î»/Î»â²,1u (BË2u +Î»â²Î»BË1Î»2(tâs) â BË2Î»â²2(tâs), BË2u) du)]).Because we have assumed λ, Î»â² > (tâs)â1/2, the upper bounds of integration in the integralsare greater than 1. We now apply the Markov property for BËi at time u = 1. We collectthe portions of the integrals from the second and third lines on the interval [0, 1], leavingthe integrals from 1 to Î»2(tâ s) and Î»â²2(tâ s). Conditional on BËi1, the Brownian motionsin the integrandsâ arguments are Brownian motions with initial position BËi1. If we denotethese by BËiu (in which case, essentially, BËiu = BËiu+1), we obtainÎÎ»,Î»â²(s) =EB0(EBË1, BË2(0,0)[EBË1,BË2(BË11 ,BË21)(Î¨Î»,Î»â²B,s (Î»â1BË1Î»2(tâs)â1, Î»â²â1BË2Î»â²2(tâs)â1)Ã exp(ââ« 10V 1,Î»â²/Î»u (BË1u, BË1u +Î»Î»â²BË2Î»â²2(tâs)â1 â BË1Î»2(tâs)â1)+ V Î»/Î»â²,1u (BË2u +Î»â²Î»BË1Î»2(tâs)â1 â BË2Î»â²2(tâs)â1, BË2u) du)Ã exp(ââ« Î»2(tâs)â10V1,Î»â²/Î»u+1 (BË1u, BË1u +Î»Î»â²BË2Î»â²2(tâs)â1 â BË1Î»2(tâs)â1) du)Ã exp(ââ« Î»â²2(tâs)â10VÎ»/Î»â²,1u+1 (BË2u +Î»â²Î»BË1Î»2(tâs)â1 â BË2Î»â²2(tâs)â1, BË2u) du))]).(2.5.13)Recall that if a process Y is defined byYr = eâr/2Berâ1where B is a standard Brownian motion, then Y is a standard one-dimensional Ornstein-Uhlenbeck process with Y0 = B0. For i = 1, 2 we let Yir = eâr/2BËierâ1. Recall thatT1 = log(Î»2(tâ s)) and T2 = log(Î»â²2(tâ s)). We therefore have thatBË1Î»2(tâs)â1 = eT1/2Y 1T1 , BË2Î»â²2(tâs)â1 = eT2/2Y 2T2 .90Expressing λ and Î»â² in terms of T1, T2 shows thatÎ»Î»â²BË2Î»2(tâs)â1 = eT1/2Y 2T2 ,Î»â²Î»BË1Î»â²2(tâs)â1 = eT2/2Y 1T1 .Likewise, we express the argument of Î¨Î»,Î»â²B,s in terms of Yi and Ti. We substitute u = erâ 1and apply the above in (2.5.13) to obtainÎÎ»,Î»â²(s) = EB0(EBË1, BË2(0,0)[EY1,Y 2(BË11 ,BË21)(Î¨Î»,Î»â²B,s (âtâ s Y 1T1 ,âtâ s Y 2T2)Ã exp(ââ« 10V 1,Î»â²/Î»u (BË1u, BË1u + eT1/2(Y 2T2 â Y 1T1)) + V Î»/Î»â²,1u (BË2u + eT2/2(Y 1T1 â Y 2T2), BË2u) du)Ã exp(ââ« T10erV1,Î»â²/Î»er (er/2Y 1r , er/2Y 1r + eT1/2(Y 2T2 â Y 1T1)) dr)Ã exp(ââ« T20erVÎ»/Î»â²,1er (er/2Y 2r + eT2/2(Y 1T1 â Y 2T2)), er/2Y 2r ) dr))]).We now apply (2.3.23) and (2.5.12) in the third and fourth lines. In the third line this giveserV1,Î»â²/Î»er (er/2Y 1r , er/2Y 1r + eT1/2(Y 2T2 â Y 1T1))= Ver/2,er/2Î»â²/Î»1 (Y1r , Y1r + e(T1âr)/2(Y 2T2 â Y 1T1))= HÎ»â²/Î»er (Y1r , Y1r + e(T1âr)/2(Y 2T2 â Y 1T1)),and similar in the fourth. Noting that V c,dt (a, b) = Vd,ct (b, a), we have obtained the desiredexpression.We now obtain an upper bound for ÎÎ»,Î»â²(s) and show that the contribution to N0((Lλt ÃLÎ»â²t )(h)) from the integral over [tâ Ç«, t] vanishes as (Ç«, Î»â²)â (0,â).Lemma 2.5.4. Suppose λ2t â¥ 1, and let h : R2 â R be bounded and measurable. There isa constant C2.5.4 > 0 such that the following hold.(a) For all Î»â² > (tâ s)â1/2,(Î»Î»â²)2Î»0â£â£ÎÎ»,Î»â²(s)â£â£ â¤ C2.5.4âhââtâÎ»0(tâ s)âÎ»0 .(b) For 0 < Ç« < t,(Î»Î»â²)2Î»0â« ttâÇ«â£â£ÎÎ»,Î»â²(s)â£â£ ds â¤ C2.5.4âhââtâÎ»0(Ç«1âÎ»0 + Î»â²â2(1âÎ»0)).Proof. To begin we use |h| â¤ âhââ and apply monotonicity (Proposition 2.3.1(a)), i.e.91V λ(x), V Î»â²(y) â¤ V Î»,Î»â²(x, y), to obtain(Î»Î»â²)2Î»0â£â£ÎÎ»,Î»â²(s)â£â£â¤ âhââ(Î»Î»â²)2Î»0EB0(EB1, B2(0,0)[exp(ââ« s0V Î»tâu(B1tâs +Bs âBu) du)Ã exp(ââ« tâs0V λr (B1r ) dr)exp(ââ« tâs0V Î»â²r (B2r ) dr)])= âhââ(Î»Î»â²)2Î»0EB10(exp(ââ« t0V λu (B1u) du))EB20(exp(ââ« tâs0V Î»â²r (B2r ) dr)),where the final line follows from a time reversal of B and concatenating the time-reversedB with B1. Applying (2.3.6) twice and changing the time variable, the above is equal toâhââ(Î»Î»â²)2Î»0EB10(exp(ââ« λ2t0V 1u (Î»B1Î»â2u) du))ÃEB20(exp(ââ« Î»â²2(tâs)0V 1u (Î»â²B2Î»â²â2u) du)).The rescaled Brownian motions in the above are themselves standard Brownian motionswhich we will denote by BË1, BË2. We next let er = u in both integrals and apply (2.3.6) tosee that the above equalsâhââ(Î»Î»â²)2Î»0EBË10(exp(ââ« log(Î»2t)ââV er/21 (eâr/2BË1er) dr))Ã EBË20(exp(ââ« log(Î»â²2(tâs))ââV er/21 (eâr/2BË2er) dr))dsâ¤ âhââ(Î»Î»â²)2Î»0EY 1m(exp(ââ« log(λ2t)0V er/21 (Y1r ) dr))Ã EY 2(exp(ââ« log(Î»â²2(tâs))ââV er/21 (Y2r ) dr)), (2.5.14)where Y ir = eâr/2BËier , which makes Y iu a stationary Ornstein-Uhlenbeck process for u â R,and we recall our assumption that λ2t â¥ 1. We condition on the value of Y 10 , which hasdistribution m.We first use the above to prove (a). Assuming that Î»â² > (tâ s)â1/2, the upper endpoint92of the second integral is positive, so by (2.5.14) we have(Î»Î»â²)2Î»0â£â£ÎÎ»,Î»â²(s)â£â£ â¤ âhââ(Î»Î»â²)2Î»0EY 1m(exp(ââ« log(λ2t)0V er/21 (Y1r ) dr))ÃEY 2m(exp(ââ« log(Î»â²2(tâs))0V er/21 (Y2r ) dr)), (2.5.15)where we have also conditioned on Y 20 . In order to approximate the expectations above withsurvival probabilities for killed Ornstein-Uhlenbeck processes, we add and subtract F (Y iu)in the integrals. Recalling the definition of ZT (Y ) from (2.3.16), we define Z1T (Y1), Z2T (Y2)in the same way. Thus (2.5.15) is equal toâhââ(Î»Î»â²)2Î»0EY 1m(Z1log(λ2t)(Y1) exp(ââ« log(λ2t)0F (Y 1r ) du))ÃEY 2m(Z2log(Î»â²2(tâs))(Y2) exp(ââ« log(Î»â²2(tâs))0F (Y 2r ) dr))â¤âhââ(Î»Î»â²)2Î»0CZEY 1m(exp(ââ« log(λ2t)0F (Y 1r ) du))Ã CZEY 2m(exp(ââ« log(Î»â²2(tâs))0F (Y 2r ) dr))ds=Câhââ(Î»Î»â²)2Î»0P Y 1m (ÏF > log(λ2t))P Y2m (ÏF > log(Î»â²2(tâ s))). (2.5.16)In the first inequality we have used (2.3.18) twice, and the second equality follows byrecognizing the expectations as survival probabilities of killed Ornstein-Uhlenbeck processeskilled at rate F (Y ir ). By (2.2.9), we haveP Y1m (ÏF > log(λ2t)) â¤ CtâÎ»0Î»â2Î»0 , P Y 2m (ÏF > log(Î»â²2(tâ s))) â¤ C(tâ s)âÎ»0Î»â²â2Î»0 .Using the above in (2.5.16), which is an upper bound for (Î»Î»â²)2Î»0â£â£ÎÎ»,Î»â²(s)â£â£, proves (a).We now show (b). Let 0 < Ç« < t. Using (2.5.14) we obtain that(Î»Î»â²)2Î»0â« ttâÇ«â£â£ÎÎ»,Î»â²(s)â£â£ ds â¤ âhââ(Î»Î»â²)2Î»0EY 1m(exp(ââ« log(λ2t)0V er/21 (Y1r ) dr))Ãâ« ttâÇ«EY2(exp(ââ« log(Î»â²2(tâs))ââV er/21 (Y2r ) dr))ds. (2.5.17)We can approximate the first expectation with the survival probability of Y 1, just as we didin the proof of (a), and bound it above by CÎ»â2Î»0tâÎ»0 . Furthermore, by the proof of part(a), we know that when Î»â² > (t â s)â1/2 the expectation in the integral above is boundedabove by C(Î»â²)â2Î»0(tâ s)âÎ»0 . When this is not the case we bound it above by 1. Thus the93right hand side of (2.5.17) is bounded above byCâhââtâÎ»0[1(Î»â² â¥ Ç«â1/2)â« tâÎ»â²â2tâÇ«(tâ s)âÎ»0 ds+ (Î»â²)2Î»0â« ttâÎ»â²â2EY2(exp(ââ« log(Î»â²2(tâs))ââV er/21 (Y2r ) dr))ds]â¤ CâhââtâÎ»0[Ç«1âÎ»0 + Î»â²â2(1âÎ»0)].The result now follows.2.5.2 ConvergenceWe now show that the expressions for N0((Lλt Ã LÎ»â²t )(h) obtained in the previous sectionconverge as λ, Î»â² â â. We do so by computing the limit explicitly. Let h : R2 â R bebounded and measurable. Clearly we may assume without loss of generality that h â¥ 0.We recall from (2.5.10) and Proposition 2.5.1 thatN0((Lλt Ã LÎ»â²t )(h)) =â« t0(Î»Î»â²)2Î»0ÎÎ»,Î»â²(s) ds,where h â¥ 0 implies ÎÎ»,Î»â²(s) â¥ 0. Our strategy is to compute the limit of (Î»Î»â²)2Î»0ÎÎ»,Î»â²(s)as λ, Î»â² â â and pass the limit through the integral. However, the scaling we use cannotbe done uniformly in s. In order to handle this and the singularity at s = t, we fix Ç« > 0and analyse the integral on [tâ Ç«, t] separately. We haveN0((Lλt Ã LÎ»â²t )(h)) =â« tâÇ«0(Î»Î»â²)2Î»0ÎÎ»,Î»â²(s) ds + (Î»Î»â²)2Î»0â« ttâÇ«ÎÎ»,Î»â²(s) ds. (2.5.18)By Lemma 2.5.4(b), the limit superior of the absolute value of the second term as Î»â² ââis bounded above by CâhââtâÎ»0Ç«1âÎ»0 . Hence, iflimÎ»,Î»â²âââ« tâÇ«0(Î»Î»â²)2Î»0ÎÎ»,Î»â²(s) dsexists for all Ç« > 0, then by the Cauchy condition limÎ»,Î»â²ââN0((LÎ»t Ã LÎ»â²t (h)) exists and isthe limit of the above as Ç« â 0. Thus it suffices to fix Ç« > 0 and establish the convergenceof, and find the limit of, the first term of (2.5.18), first as λ, Î»â² ââ and then as Ç« â 0. ByLemma 2.5.4(a), we have(Î»Î»â²)2Î»0 |ÎÎ»,Î»â²(s)| â¤ g(s) for all s â [0, tâ Ç«]94for all λ, Î»â² > Ç«â1/2 for a function g(s) â¥ 0 satisfying â« tâÇ«0 g(s)ds < â. It follows that, if(Î»Î»â²)2Î»0ÎÎ»,Î»â²(s) converges as λ, Î»â² ââ, then Dominated Convergence implies thatlimÎ»,Î»â²ââN0((LÎ» Ã LÎ»â²t )(h)) = limÇ«â0+limÎ»,Î»â²âââ« tâÇ«0(Î»Î»â²)2Î»0ÎÎ»,Î»â²(s) ds= limÇ«â0+â« tâÇ«0limÎ»,Î»â²ââ(Î»Î»â²)2Î»0ÎÎ»,Î»â²(s) ds, (2.5.19)and so it suffices to find the limit of (Î»Î»â²)2Î»0ÎÎ»,Î»â²(s) as λ, Î»â² ââ.Let s â (0, t) and assume λ, Î»â² > (tâ s)â1/2. By Lemma 2.5.3,(Î»Î»â²)2Î»0ÎÎ»,Î»â²(s) = (Î»Î»â²)2Î»0EB0(EB1,B2(0,0)(EY1,Y 2B11 ,B21[Î¨Î»,Î»â²B,s (âtâ s Y 1T1 ,âtâ s Y 2T2)Ã exp(ââ« 10V 1,Î»â²/Î»u (B1u, B1u + eT1/2(Y 2T2 â Y 1T1)) + V 1,Î»/Î»â²u (B2u, B2u + eT2/2(Y 1T1 â Y 2T2))du)Ã exp(ââ« T10HÎ»â²/Î»eu (Y1u , Y1u + eT1âu2 (Y 2T2 â Y 1T1)) du)Ã exp(ââ« T20HÎ»/Î»â²eu (Y2u , Y2u + eT2âu2 (Y 1T1 â Y 2T2)) du)))], (2.5.20)where T1 = T1(s) = log(λ2(t â s)), T2 = T2(s) = log(Î»â²2(t â s)). Inside the integral in thethird term we add and subtract F (Y iu) and decompose as followsexp(ââ« T10H1eu(Y1u , Y1u + eTâu2 (Y 2T2 â Y 1T1)) du)= exp(ââ« T10F (Y 1u ) du)exp(â« T10F (Y 1u )âHÎ»â²/Î»eu (Y1u , Y1u + eT1âu2 (Y 2T2 â Y 1T1)) du).We do the same to the fourth term with the obvious changes of indices. The first term inthe above is the probability that the Ornstein-Uhlenbeck process Y 1 with killing function Fsurvives until time T1. We extract a similar term from the symmetric term correspondingto Y 2 and T2. Weighting the expectation of a functional with this survival probability isequivalent to restricting the expectation to the event that the process survives; in our case,we restrict to the event that Y 1 and Y 2 survive until T1 and T2, respectively. Thus (2.5.20)95is equal to(Î»Î»â²)2Î»0EB0(EB1,B2(0,0)(EY1,Y 2B11 ,B21[Î¨Î»,Î»â²B,s (âtâ s Y 1T1 ,âtâ s Y 2T2)Ã exp(ââ« 10V 1,Î»â²/Î»u (B1u, B1u + eT1/2(Y 2T2 â Y 1T1)) + V 1,Î»/Î»â²u (B2u, B2u + eT2/2(Y 1T1 â Y 2T2))du)Ã exp(â« T10F (Y 1u )âHÎ»â²/Î»eu (Y1u , Y1u + eT1âu2 (Y 2T2 â Y 1T1)) du)Ã exp(â« T20F (Y 2u )âHÎ»/Î»â²eu (Y2u , Y2u + eT2âu2 (Y 1T1 â Y 2T2)) du)1(Ï1 > T1)1(Ï2 > T2)])),(2.5.21)where Ïi = ÏFi is the lifetime of the killed process Yi. Recall the transition density qt(Â·, Â·)(with respect to m) of the killed diffusion. We condition on the endpoints Y iTi = zi (recallfrom Lemma 2.2.3(a) that the regular conditional distributions exist for all zi â R) andintegrate against qTi(Â·, zi) dm(zi) to obtain that (2.5.21) is equal to(Î»Î»â²)2Î»0EB0(EB1,B2(0,0)( â«â«Î¨Î»,Î»â²B,s (âtâ sz1,âtâ sz2)Ã exp(ââ« 10V 1,Î»â²/Î»u (B1u, B1u + eT1/2(z2 â z1)) + V 1,Î»/Î»â²u (B2u, B2u + eT2/2(z1 â z2)) du)Ã EY 1B11(exp(â« T10F (Y 1u )âHÎ»â²/Î»eu (Y1u , Y1u + eT1âu2 (z2 â z1)) du)â£â£â£â£ Ï1 > T1, Y 1T1 = z1)Ã EY 2B21(exp(â« T20F (Y 2u )âHÎ»/Î»â²eu (Y2u , Y2u + eT2âu2 (z1 â z2)) du)â£â£â£â£ Ï2 > T2, Y 2T2 = z2)Ã qT1(B11 , z1) qT2(B21 , z2) dm(z1) dm(z2)))=:â«â«(Î»Î»â²)2Î»0[EB0(EB1,B2(0,0)(G(Î», Î»â², s, B,B1, B2, z1, z2)Ã qT1(B11 , z1) qT2(B21 , z2))) ]dm(z1) dm(z2). (2.5.22)The function G is defined implicitly. The conditional probabilities that appear are the samethat are defined in Section 2.2, in particular Lemma 2.2.3. We have used that the termsin the third and fourth lines are independent conditional on the endpoints. Hereafter, Y 1and Y 2, and their respective laws, refer to killed Ornstein-Uhlenbeck processes with killingfunction F . Furthermore, after this point we will suppress the conditioning on Ïi > Ti, asit is implicit in the conditioning Y iTi = zi that Ïi > Ti.96We introduce notation for the terms appearing in G(λ, Î»â², s, B,B1, B2, z1, z2). We defineQ(λ, Î»â², B1, B2, z1, z2) (2.5.23):= exp(ââ« 10V 1,Î»â²/Î»u (B1u, B1u + eT1/2(z2 â z1)) + V 1,Î»/Î»â²u (B2u, B2u + eT2/2(z1 â z2)) du),andZË1T1 = ZË1T1(Y1, z1, z2, Î»â²/Î»):= exp(â« T10F (Y 1u )âHÎ»â²/Î»eu (Y1u , Y1u + eT1âu2 (z2 â z1)) du), (2.5.24)ZË2T2 = ZË2T2(Y2, z2, z1, Î»/Î»â²):= exp(â« T20F (Y 2u )âHÎ»/Î»â²eu (Y2u , Y2u + eT2âu2 (z1 â z2)) du). (2.5.25)Recall that Î¨Î»,Î»â²B,s (âtâ sz1,âtâ sz2) was defined in (2.5.11). From (2.5.22) we haveG(λ, Î»â², s, B,B1, B2, z1, z2) (2.5.26)= Î¨Î»,Î»â²B,s (âtâ sz1,âtâ sz2)Q(λ, Î»â², B1, B2, z1, z2)EY 1B11 (ZË1T1â£â£Y 1T1 = z1)EY 2B21 (ZË2T2 â£â£Y 2T2 = z2).We note that ZË1T1 and ZË2T2are perturbations of the corresponding ZiTi terms. In particular,we defined ZiTi byZiTi(Yi) = exp(â« Ti0F (Y iu)â V eu/21 (Yiu) du). (2.5.27)By Proposition 2.3.1(a) and (2.5.12), we have that Hceu(x, y) â¥ V eu/21 (x), and henceZËiTi â¤ ZiTi(Y i) â¤ CZ , (2.5.28)where the second inequality is by (2.3.18). Using Q(λ, Î»â², B1, B2, z1, z2) â¤ 1 and |Î¨Î»,Î»â²B,s | â¤âhââ, both of which are obvious from these termsâ definitions, we therefore obtain that fora constant C1 > 0,|G(λ, Î»â², s, B,B1, B2, z1, z2)| â¤ C1 (2.5.29)uniformly in its arguments. We now define Î(Î», Î»â², s, B,B1, B2, z1, z2) as the function inthe square-bracketed term in (2.5.22) multiplied by the scaling factor (Î»Î»â²)2Î»0 . That is,Î(Î», Î»â², s, B,B1, B2, z1, z2):= G(λ, Î»â², s, B,B11 , B21 , z1, z2) (Î»Î»â²)2Î»0 qT1(B11 , z1) qT2(B21 , z2). (2.5.30)97Note from (2.5.22) thatÎÎ»,Î»â²(s) =â«â«EB0 (EB1,B2(0,0) (Î(Î», Î»â², s, B,B1, B2, z1, z2))) dm(z1) dm(z2). (2.5.31)Recall that T1 = log(λ2(t â s)) and T2 = log(Î»â²2(t â s)). Taking sâ(1/8) as in Theo-rem 2.2.1(c), we note that if λ, Î»â² > esâ/2(tâ s)â1/2, then T1, T2 â¥ sâ(1/8). We define λ¯(s)asλ¯(s) :=[esâ(1/8)/2(tâ s)â1/2] â¨ 1 (2.5.32)and Ï(s) byÏ(s) = log(Î»¯(s)2(tâ s)). (2.5.33)Applying (2.2.2) with δ = 1/8, we obtainqT1(b1, z1) qT2(b2, z2) â¤ C(tâ s)â2Î»0(Î»Î»â²)â2Î»0e1/8(b21+b22+z21+z22)for all T1, T2 > Ï(s) (equivalently, λ, Î»â² > λ¯(s)). Using the above and (2.5.29), we obtain|Î(Î», Î»â², s, B,B1, B2, z1, z2)|â¤ C (tâ s)â2Î»0 exp ([(B11)2 + (B21)2 + z21 + z22] /8) for all λ, Î»â² > λ¯(s). (2.5.34)Since Bi1 â¼ m, (2.5.34) implies that Î has a (uniform in λ, Î»â² > λ¯(s)) upper boundwhich is integrable with respect to dPB0 dPB10 dPB20 dm(z1) dm(z2). From (2.5.31), thisimplies that (Î»Î»â²)2Î»0ÎÎ»,Î»â²(s) is bounded for λ, Î»â² > λ¯(s) (for fixed s < t). Moreover,if limÎ»,Î»â²ââÎ(Î», Î»â², s, B,B1, B2, z1, z2) exists for PB0 â PB1,B2(0,0) -a.a. Ï and Lebesgue-a.a.z1, z2 â R, then by (2.5.31) and Dominated Convergence (using (2.5.34)), we havelimÎ»,Î»â²ââ(Î»Î»â²)2Î»0ÎÎ»,Î»â²(s)=â«â«EB0 (EB1,B2(0,0) [ limÎ»,Î»â²ââÎ(Î», Î»â², s, B,B1, B2, z1, z2)]) dm(z1) dm(z2). (2.5.35)In view of (2.5.19), the above implies the following:If limÎ»,Î»â²ââÎ(Î», Î»â², s, B,B1, B2, z2, z2) exists PB0 â PB1,B2(0,0) -a.s. for a.e. z1, z2 â R,then limÎ»,Î»â²ââN0((LÎ»t Ã LÎ»â²t )(h))=â« t0[ â«â«EB0 (EB1,B2(0,0) [ limÎ»,Î»â²ââÎ(Î», Î»â², s, B,B1, B2, z1, z2)]) dm(z1) dm(z2)]ds.(2.5.36)98As h â¥ 0, and hence ÎÎ»,Î»â²(s) â¥ 0, the right hand side of the above is equal to the lastexpression of (2.5.19) (provided Î converges) by Monotone Convergence. Thus it sufficesto compute the limit of Î(Î», Î»â², s, B,B1, B2, z2, z2) as λ, Î»â² â â. As we only need to findthe limit a.e. in (z1, z2), we will hereafter assume that z1 6= z2. We also take this opportunityto reiterate our assumptions about λ and Î»â². Originally we assumed λ, Î»â² > (tâ s)â1/2; inview of the above, we augment the assumption to λ, Î»â² > λ¯(s), or equivalently, T1, T2 > Ï(s)(see (2.5.32) and (2.5.33)). This implies that λ, Î»â² > 1 and T1, T2 > sâ(1/8).Î is the product of the function G and the rescaled transition densities, that is,λ2λ0qT1(B11 , z1) and Î»â²2Î»0qT2(B21 , z2). We will show that both of these approach finite limitsas λ, Î»â² ââ. First, let us handle the transition densities. By Lemma 2.2.2,limTiââeÎ»0TiqTi(Bi1, zi) = Ï0(Bi1)Ï0(zi)for i = 1, 2. Using the definitions of T1 and T2 (e.g. T1 = log(Î»2(tâ s))), we readily obtainfrom the above thatλ2λ0qT1(B11 , z1)â (tâ s)âÎ»0Ï0(B11)Ï0(z1) as Î»ââ, andÎ»â²2Î»0qT2(B21 , z2)â (tâ s)âÎ»0Ï0(B21)Ï0(z2) as Î»â² ââ (2.5.37)for all B11 , B21 , z1, z2 â R.We now compute the limit of G, whose definition we recall from (2.5.26). We proceedby computing the limit of each constituent term. The analysis is most technical for theconditional expectations of ZËiTi , for i = 1, 2, which were defined in (2.5.24) and (2.5.25).These two quantities are essentially the same with different parameters. To avoid excessiveand cumbersome notation we define a variable ZËT . Let λ, Î»â² > 0 and T = log(λ2(t â s)),and x, z2, z2 â R. For a killed Ornstein-Uhlenbeck process Y , we defineZËT = ZËT (Y, z1, z2, Î»â²/Î») := exp(â« T0F (Yu)âHÎ»â²/Î»eu (Yu, Yu+ eTâu2 (z2 â z1)) du). (2.5.38)In order to characterize the limit of the conditional expectation of ZËT (as in (2.5.26)), weintroduce a quantity WS(Y, z). For S > 0 and z â R, we defineWS(Y, z) = exp(â« S0F (Yu)â F2(Yu, Yu â eu/2(z â Y0)) du), (2.5.39)where we recall from (2.3.24) that F2(a, b) = Vâ,â1 (a, b). By Proposition 2.3.1(a), F (a) â¤F2(a, b) for all a, b â R, so the integrand is non-positive and hence WS(Y, z) â¤ 1 and isnon-increasing in S. Since it is bounded below by 0, we can define its monotone limit as99Wâ(Y, z) = limSââWS(Y, z) â¤ 1.Lemma 2.5.5. Let x, z1, z2 â R such that z1 6= z2. ThenlimÎ»,Î»â²ââEYx(ZËT (Y, z1, z2, Î»â²/Î»)â£â£YT = z1) = EY,âx (Zâ(Y ))EY,âz1 (Wâ(Y, z2)).Recall that EY,âx is the expectation under the law of the killed process Y with Y0 = xconditioned to survive for all time, as defined in Theorem 2.2.1(e). ZT (Y ) is as defined in(2.3.16) and we recall from (2.3.17) and (2.3.18) that Zâ(Y ) = limTââZT (Y ) exists andis bounded by CZ .Heuristically, the Zâ term in the limiting expression in Lemma 2.5.5 comes from theearly (small u) part of the integral in ZËT , and the Wâ term comes from the tail part (i.e.u near T ), and these two contributions are asymptotically independent. Since the time atwhich we condition is T and T goes to infinity, in the limit the expectations are computedunder the measure of the process conditioned to survive forever.Section 2.6 is devoted to the proof of Lemma 2.5.5. For now, we carry on with the proofof Theorem 2.1.4. Returning to ZË1T1 and ZË2T2, it follows from Lemma 2.5.5 thatFor all B11 , B21 â R and all z1, z2 â R such that z1 6= z2,limÎ»,Î»â²ââEY1B11(ZË1T1(Y1, z1, z2, Î»â²/Î»)â£â£ Y 1T1 = z1) = EY,âB11 (Zâ(Y ))EY,âz1 (Wâ(Y, z2)), andlimÎ»,Î»â²ââEY2B21(ZË2T2(Y2, z2, z1, Î»/Î»â²)â£â£Y 2T2 = z2) = EY,âB21 (Zâ(Y ))EY,âz2 (Wâ(Y, z1)). (2.5.40)To find the limit of G it remains to identify the limits of Î¨Î»,Î»â²B,s (âtâ sz1,âtâ sz2) andQ(λ1, λ2, B1, B2, z1, z2). From (2.5.11) we recall that the former prelimit is defined asÎ¨Î»,Î»â²B,s (âtâ sz1,âtâ sz2) = h(âtâ sz1 +Bs,âtâ sz2 +Bs)Ã exp(ââ« s0V Î»,Î»â²tâu (âtâ sz1 +Bs âBu,âtâ sz2 +Bs âBu) du).For all z1, z2 â R and all Brownian paths (Bu, u â [0, s]), the obvious limit of the aboveas λ, Î»â² â â is obtained by replacing V Î»,Î»â²tâu with Vâ,âtâu . By monotonicity (in λ, Î»â²) of theintegral and continuity of the exponential we can take the limit inside. Denoting the limitby Î¨â,âB,s (âtâ sz1,âtâ sz2), we havelimÎ»,Î»â²ââÎ¨Î»,Î»â²B,s (âtâ sz1,âtâ sz2) = Î¨â,âB,s (âtâ sz1,âtâ sz2). (2.5.41)100This leaves Q(λ, Î»â², B1, B2, z1, z2), which we recall from (2.5.23) is defined byQ(λ, Î»â², B1, B2, z1, z2)= exp(ââ« 10V 1,Î»â²/Î»u (B1u, B1u + eT1/2(z2 â z1)) + V 1,Î»/Î»â²(B2u, B2u + eT2/2(z1 â z2)) du).The integrand is the sum of two terms that are very similar; for now we restrict our attentionto the first. In particular, we will show thatlimT1ââsupÎ»â²>(tâs)â1/2â£â£â£â£ exp(ââ« 10V 1,Î»â²/Î»u (B1u, B1u + eT1/2(z2 â z1)) du)â exp(ââ« 10V 1u (B1u) du)â£â£â£â£ = 0. (2.5.42)We claim that since the second argument of the integrand goes to infinity, asymptoticallythe function resembles V 1u (B1u). To see this use both parts of Proposition 2.3.1 to concludethat0 â¤[V 1,cu (B1u, B1u + eT1/2(z2 â z1))â V 1u (B1u)]â¤ Vâu (B1u + eT1/2(z2 â z1))for all c > 0. PB0 -a.s., there is a constant R(Ï) > 0 such that |B1u(Ï)| â¤ R(Ï) for allu â [0, s]. Provided z1 6= z2, for sufficiently large λ, eT1/2|z2 â z1| â¥ 2R. Then for λsufficiently large and λ, Î»â² > Î»¯(s),â£â£â£â£ exp(ââ« 10V 1,Î»â²/Î»u (B1u, B1u + eT1/2(z2 â z1)) du)â exp(ââ« 10V 1u (B1u) du)â£â£â£â£â¤â« 10Vâu (eT1/2(z2 â z1)âR) du.The integrand is bounded above by Vâu (R). Since Vâu (R) = uâ1F (uâ1/2R), (from (2.3.13))by (2.3.14)(iii) we have Vâu (R) â¤ cuâ3/2Reâuâ1R/2, which is bounded on [0, 1]. We take Î»ââ and apply Dominated Convergence; since Vâu (y) = uâ1F (uâ1/2y), applying (2.3.14)(iii)again gives that lim|y|ââ Vâu (y) = 0, and hence limit of the above as λ â â (i.e. asT1 â â) is zero. Thus (2.5.42) holds. We handle the second term in the integral inQ(λ, Î»â², B1, B2, z1, z2) in an identical fashion, now with the roles of λ and Î»â² reversed,thereby establishing thatFor all z1, z2 â R such that z1 6= z2, dPB10 dPB20 -a.s., (2.5.43)limÎ»,Î»â²ââQ(Î», Î»â², B1, B2, z1, z2) = exp(ââ« 10V 1u (B1u) du)exp(ââ« 10V 1u (B2u) du).101We have therefore found the limit of G and hence of Î. In particular, recall the definitions(2.5.26) and (2.5.30). From (2.5.37), (2.5.40), (2.5.41) and (2.5.43), we have shown thatdPB0 dPB10 dPB20 -a.s., for all z1, z2 â R such that z1 6= z2,limÎ»,Î»â²ââÎ(Î», Î»â², s, B,B1, B2, z1, z2) = (tâ s)â2Î»0Î¨â,âB,s (âtâ sz1,âtâ sz2)Ã EY,âz1 (Wâ(Y, z2))EY,âz2 (Wâ(Y, z1))EY,âB11 (Zâ(Y ))EY,âB21(Zâ(Y ))Ã exp(â« 10âV 1u (B1u)â V 1u (B2u) du)Ï0(B11)Ï0(B21)Ï0(z1)Ï0(z2). (2.5.44)Thus by (2.5.36), limÎ»,Î»â²ââN0((LÎ»t Ã LÎ»â²t )(h)) exists and satisfieslimÎ»,Î»â²ââN0((LÎ»t Ã LÎ»â²t )(h))=â« t0[ â«â«EB0 (EB1,B2(0,0) [ limÎ»,Î»â²ââÎ(Î», Î»â², s, B,B1, B2, z1, z2)]) dm(z1) dm(z2)]ds. (2.5.45)To obtain the desired expression, we note that the terms in (2.5.44) that depend on B1 andB2 can be collected in a constant. In particular, we define a constant C2.1.4 > 0 byC22.1.4 = EB1,B2(0,0)(exp(ââ« 10V 1u (B1u) + V1u (B2u) du)Ã EY,âB11(Zâ(Y ))EY,âB21(Zâ(Y ))Ï0(B11)Ï0(B21))=[EB0(exp(ââ« 10V 1u (Bu) du)EY,âB1(Zâ(Y ))Ï0(B1))]2. (2.5.46)We also define a function Ï(Â·, Â·) byÏ(z1, z2) = EY,âz1 (Wâ(Y, z2))EY,âz2 (Wâ(Y, z1)). (2.5.47)It is clear that Ï(Â·, Â·) is jointly continuous and bounded by 1 from the definition ofWâ(Y, z).Thus by (2.5.44),EB1,B2(0,0)[limÎ»,Î»â²ââÎ(Î», Î»â², s, B,B1, B2, z1, z2)]= C22.1.4(tâ s)â2Î»0 Î¨â,âB,s (âtâ sz1,âtâ sz2) Ï(z1, z2)Ï0(z1)Ï0(z2).Substituting the above into (2.5.45) completes the proof of Theorem 2.1.4.1022.6 Proof of Lemma 2.5.5Recalling the statement of the lemma, we will show thatlimÎ»,Î»â²ââEYx(ZËT (Y, z1, z2, Î»â²/Î»)â£â£ YT = z1) = EY,âx (Zâ(Y ))EY,âz1 (Wâ(Y, z2)) (2.6.1)for all x, z1, z2 â R such that z1 6= z2, where the law of Y on the left hand side is that of akilled Ornstein-Uhlenbeck process with killing function F , with Y0 = x. Recall that EY,âxis the expectation under the law of the killed process Y with Y0 = x conditioned to survivefor all time. For convenience, we now recall the definitions of the quantities above: from(2.3.16), (2.5.38) and (2.5.39),ZËT (Y, z1, z2, Î»â²/Î») = exp(â« T0F (Yu)âHÎ»â²/Î»eu (Yu, Yu + eTâu2 (z2 â z1)) du),WS(Y, z) = exp(â« S0F (Yu)â F2(Yu, Yu â eu/2(z â Y0)) du),ZT (Y ) = exp(â« T0F (Yu)â V eu/21 (Yu) du).As we have previously discussed, Zâ(Y ) and Wâ(Y, z) are the respective limits of ZT (Y )and WS(Y, z) as T, S â â, both of which exist. Recall from (2.3.18) that Zâ(Y ) â¤ CZ .Because z1 and z2 are fixed, we will hereafter suppress the dependence of ZËT (Y, z1, z2, Î»â²/Î»)on z1 and z2 and simply write ZËT (Y, Î»â²/Î»). Finally, we recall our working assumption thatλ, Î»â² > λ¯(s), (see (2.5.32)) which implies that λ, Î»â² > 1 and T > sâ(1/8).Let 0 < K < T/2. We apply the Markov property to EYx (ZËT (Y, Î»â²/Î»)â£â£YT = z1) at timesK and T âK and expand in terms of the joint density of (YK , YTâK). As in (2.2.11), thejoint density of (YK , YTâK) at (w, y) with respect to mÃm under P Yx (Y â · |YT = z1) isqK(x,w)qTâ2K(w, y)qK(y, z1)qT (x, z1).Thus we obtain the following:EYx(ZËT (Y, Î»â²/Î»)â£â£ YT = z1)= EYx(exp(â« T0F (Yu)âHÎ»â²/Î»eu (Yu, Yu + eTâu2 (z2 â z1)) du)â£â£â£â£YT = z1)103=â«â«EYx(exp(â« K0F (Yu)âHÎ»â²/Î»eu (Yu, Yu + eTâu2 (z2 â z1)) du)â£â£â£â£YK = w)Ã EYw(exp(â« Tâ2K0F (Yu)âHÎ»â²/Î»eK+u(Yu, Yu + eTâKâu2 (z2 â z1)) du)â£â£â£â£YTâ2K = y)Ã EYy(exp(â« K0F (Yu)âHÎ»â²/Î»eTâK+u(Yu, Yu + eKâu2 (z2 â z1)) du)â£â£â£â£YK = z1)Ã qK(x,w)qTâ2K(w, y)qK(y, z1)qT (x, z1)dm(w) dm(y). (2.6.2)Denote the three conditional expectations by A1(x,w, λ, Î»â²,K), A2(w, y, λ, Î»â²,K) andA3(y, z1, λ, Î»â²,K). That is,A1(x,w, λ, Î»â²,K) (2.6.3)= EYx(exp(â« K0F (Yu)âHÎ»â²/Î»eu (Yu, Yu + eTâu2 (z2 â z1)) du)â£â£â£â£YK = w),A2(w, y, λ, Î»â²,K) (2.6.4)= EYw(exp(â« Tâ2K0F (Yu)âHÎ»â²/Î»eK+u(Yu, Yu + eTâKâu2 (z2 â z1)) du)â£â£â£â£YTâ2K = y),A3(y, z1, λ, Î»â²,K) (2.6.5)= EYy(exp(â« K0F (Yu)âHÎ»â²/Î»eTâK+u(Yu, Yu + eKâu2 (z2 â z1)) du)â£â£â£â£YK = z1).We observe that A1, A2 and A3 all depend on z1 and z2 in addition to their listed arguments,as these values appear in their integrands. Again, as z1 and z2 are fixed, we omit thisadditional dependence. Noting that the integrand is bounded above by F (Yu)â V eu/21 (Yu)in each case, from (2.3.18) we have Ai â¤ CZ for i = 1, 2, 3. In terms of the Ai, (2.6.2) canbe rewritten asEYx(ZËT (Y, Î»â²/Î»)â£â£ YT = z1) = â«â« A1(x,w, λ, Î»â²,K)A2(w, y, λ, Î»â²,K)A3(y, z1, λ, Î»â²,K)Ã qK(x,w) qTâ2K(w, y) qK(y, z1)qT (x, z1)dm(w) dm(y). (2.6.6)There are two main contributions in the Ai. The first comes from F and the first argumentof the H function, and is approximately equal to F (Yu)âV eu/21 (Yu); the second comes fromthe second argument of the H function. We will see that, asymptotically, A1 is only affectedby the first contribution and gives the Zâ(Y ) term in (2.6.1); A3 is only affected by thesecond contribution and gives the Wâ(Y, z2) term in (2.6.1). The contribution of A2 is willbe seen to be negligible. We first show that A2 is arbitrarily close to 1 as K is made large,uniformly in T sufficiently large depending on K. Define ZaT (Y, Î»â²/Î»,K) as ZËT (Y, Î»â²/Î»,K)104with A2 replaced by 1; that is,ZaT (Y, Î»â²/Î»,K) = exp(â« K0F (Yu)âHÎ»â²/Î»eu (Yu, Yu + eTâu2 (z2 â z1)) du)Ã exp(â« TTâKF (Yu)âHÎ»â²/Î»eu (Yu, Yu + eTâu2 (z2 â z1)) du). (2.6.7)As in (2.6.2) and (2.6.6), we therefore haveEYx(ZaT (Y, Î»â²/Î»,K)â£â£YT = z1)=â«â«A1(x,w, λ, Î»â²,K)A3(y, z1, λ, Î»â²,K)qK(x,w) qTâ2K(w, y) qK(y, z1)qT (x, z1)dm(w) dm(y).(2.6.8)By monotonicity (Proposition 2.3.1(a)) and (2.3.15) we haveF (Yu)âHÎ»â²/Î»eK+u(Yu, Yu + eTâKâu2 (z2 â z1)) â¤ F (Yu)â V e(K+u)/21 (Yu) â¤ Ceâ(K+u)(2Î»0â1)/2(2.6.9)uniformly in T > 2K. Integrating this over u shows that the exponent in A2 is boundedabove C â²eâ(2Î»0â1)K/2 for a constant C â², uniformly in T > 2K. We choose K large enoughso that exponent in A2 is smaller than 2. Then by (2.6.6) and (2.6.7), applying the meanvalue theorem, we haveâ£â£EYx (ZËT (Y, Î»â²/Î»)â ZaT (Y, Î»â²/Î»,K) â£â£ YT = z1)â£â£â¤ 1qT (x, z1)â«â«A1(x,w, λ, Î»â²,K)â£â£â£â£A2(w, y, λ, Î»â²,K)â 1â£â£â£â£A3(y, z1, λ, Î»â²,K)Ã qK(x,w)qTâ2K(w, y)qK(y, z1) dm(w) dm(y)â¤ e2C2ZqT (x, z1)â«â«EYw(â« Tâ2K0|F (Yu)âHÎ»â²/Î»eK+u(Yu, Yu + eTâKâu2 (z2 â z1))| duâ£â£â£â£YTâ2K = y)Ã qK(x,w)qTâ2K(w, y)qK(y, z1) dm(w) dm(y) (2.6.10)uniformly for all T > 2K, where we have also used A1A3 â¤ C2Z . The term in the absolutevalue inside the integral can be positive or negative; (2.6.9) provides an upper bound forF âHÎ»â²/Î»eK+u. To obtain a lower bound, we note that HÎ»â²/Î»eK+u(a, b) â¤ F2(a, b) â¤ F (a) + F (b)by Proposition 2.3.1 (using part (a) and then part (b)). This bound implies thatF (Yu)âHÎ»â²/Î»eK+u(Yu, Yu + eTâKâu2 (z2 â z1)) â¥ âF (Yu + eTâKâu2 (z2 â z1)). (2.6.11)Together, (2.6.9) and (2.6.11) imply that the absolute value appearing in the integral in105(2.6.10) is bounded above byCeâ(K+u)(2Î»0â1)/2 + F (Yu + eTâKâu2 (z2 â z1)).We have already noted that when integrated over u, the first term is bounded by C â²eâK(2Î»0â1)/2(uniformly in T ). The first term has no dependence on the spatial parameters w and y, so in(2.6.10) the transition densities and can be integrated and cancelled with the denominator.We get that for all T > 2K, (2.6.10) is bounded above byC â²eâK(2Î»0â1)/2 +CqT (x, z1)â«â«EYw(â« Tâ2K0F (Yu + eTâKâu2 (z2 â z1)) duâ£â£â£â£ YTâ2K = y)Ã qK(x,w)qTâ2K(w, y)qK(y, z1) dm(w) dm(y).We consider the time reversed process in the above and apply (2.2.12), which implies thatthe above is equal to, and hence for all T > 2K, (2.6.10) is bounded above byC â²eâK(2Î»0â1)/2 +CqT (x, z1)â«â«EYy(â« Tâ2K0F (Yu + eK+u2 (z2 â z1)) duâ£â£â£â£ YTâ2K = w)Ã qK(x,w)qTâ2K(w, y)qK(y, z1) dm(w) dm(y). (2.6.12)We recall the asymptotic behaviour of F from (2.3.14)(iii), i.e. that F (x) â¼ c1|x|eâx2/2 as|x| â â. This implies there is a constant c2 > 0 such thatF (x) â¤ c2(1 + |x|)eâx2/2 for all x â R. (2.6.13)In order for this to give a useful upper bound in (2.6.12), weâll need to show that theargument of F is large in absolute value. It is enough to show that |Yu| âª eK+u2 |z2 â z1|with high probability when conditioned on its endpoint. Recall that we have assumedz1 6= z2. We bound the integrand over the two cases mentioned above and exchange theintegral and expectation, which is justifiable since F is positive. We haveEYy(â« Tâ2K0F (Yu + eK+u2 (z2 â z1)) duâ£â£â£â£ YTâ2K = w)â¤ EYy(â« Tâ2K0F (eK+u4 |z2 â z1|) + F (0)1(|Yu| â¥ eK+u4 |z2 â z1|) duâ£â£â£â£ YTâ2K = w)â¤ c2â« â0(1 + eK+u4 |z2 â z1|)eâeK+u2 |z2âz1|2/2du+ F (0)â« Tâ2K0P Yy (|Yu| â¥ eK+u4 |z2 â z1|â£â£YTâ2K = w) du, (2.6.14)106where we have used (2.6.13) and the fact that F is radially decreasing. A simple substitutionshows thatc2â« â0(1 + eK+u4 |z2 â z1|)eâeK+u2 |z2âz1|2/2duâ¤ 4c1â« âeK/4|z2âz1|(1 + aâ1)eâa2/2daâ¤ Câ« âeK/4|z2âz1|eâa2/2da+C1(eK/4|z2 â z1| < 1)â« 1eK/4|z2âz1|aâ1eâa2/2daâ¤ Câ« âeK/4|z2âz1|eâa2/2daâC[log(eK/4|z2 â z1|) â§ 0]. (2.6.15)To bound the second term in (2.6.14) we expand the probability of the large excursion interms of the transition densities. There are two cases, which we handle in the followinglemma. In what follows, sâ = sâ(1/8) from Theorem 2.2.1(c).Lemma 2.6.1. Let M > 0 and w, y â R.(a) There is a constant C > 0 such that for S, u > 0 satisfying u, S â u â¥ sâ,P Yy (|Yu| â¥M |YS = w) â¤CqS(y,w)eâÎ»0Se(y2+w2)/8[eâM2/4Mâ§ 1].(b) For fixed u0 > 0 the families{P Yy (Yu â · |YS = w) : S â¥ u0, 0 â¤ u â¤ u0} and{P Yy (YSâu â · |YS = w) : S â¥ u0, 0 â¤ u â¤ u0}are tight.Proof. To see (a), we use (2.2.11) and (2.2.2) with δ = 1/8 to obtain that for u, S âu â¥ sâ,P Yy (Yu â¥Mâ£â£YS = w) = â« âMqu(y, a)qSâu(a,w)qS(y,w)dm(a)â¤c21/8qS(y,w)eâÎ»0Sâ« âMe(y2+2a2+w2)/8 dm(a)â¤ CqS(y,w)eâÎ»0Se(y2+w2)/8[eâM2/4Mâ§ 1],where the last line uses a standard upper bound on Gaussian tails and bounds the integralabove by a constant when M is small. The bound for Yu < âM is the same. The firstfamily in part (b) is tight as a consequence of Lemma 2.2.3(c). To see that the secondfamily is tight we consider the time reversal of Y and use (2.2.12), from which tightness107now also follows from Lemma 2.2.3(c).Applying Lemma 2.6.1(a), using (2.6.15) and separating the integrals depending if u, Sâu â¥ sâ or not, we have that (2.6.14) is bounded above byCâ« âeK/4|z2âz1|eâa2/2daâ C[log(eK/4|z2 â z1|) â§ 0]+CqTâ2K(y,w)eâÎ»0(Tâ2K)ey2/8ew2/8â« Tâ2Kâsâsâï£®ï£°eâeK+u2 |z2âz1|2/4eK+u4 |z2 â z1|â§ 1ï£¹ï£» du+ C(â« sâ0+â« Tâ2KTâ2Kâsâ)P Yy (|Yu| â¥ eK+u4 |z2 â z1|â£â£YTâ2K = w) du. (2.6.16)As the above is an upper bound for the expectation appearing in the second term of (2.6.12),and (2.6.12) is an upper bound for (2.6.10), we haveâ£â£EYx (ZËT (Y, Î»â²/Î»)â ZaT (Y, Î»â²/Î»,K) â£â£ YT = z1)â£â£â¤ CeâK(2Î»0â1)/2 + CqT (x, z1)â«â« [ â« âeK/4|z2âz1|eâa2/2daâ C[log(eK/4|z2 â z1|) â§ 0]+1qTâ2K(y,w)eâÎ»0(Tâ2K)ey2/8ew2/8â« Tâ2Kâsâsâï£®ï£°eâeK+u2 |z2âz1|2/4eK+u4 |z2 â z1|â§ 1ï£¹ï£» du+(â« sâ0+â« Tâ2KTâ2Kâsâ)P Yy (|Yu| â¥ eK+u4 |z2 â z1|â£â£YTâ2K = w) du]Ã qK(x,w)qTâ2K(w, y)qK(y, z1) dm(w) dm(y). (2.6.17)Note that the first two terms in the integral with respect to y and w are independent ofthese variables. We can therefore integrate them out; using the fact thatâ«â«qK(x,w)qTâ2K(w, y)qK(y, z1) dm(w) dm(y) = qT (x, z1)108(and an obvious cancellation) we obtain thatâ£â£EYx (ZËT (Y, Î»â²/Î»)â ZaT (Y, Î»â²/Î»,K) â£â£YT = z1)â£â£â¤ CeâK(2Î»0â1)/2 + Câ« âeK/4|z2âz1|eâa2/2daâ C[log(eK/4|z2 â z1|) â§ 0]+CqT (x, z1)eâÎ»0(Tâ2K)â«â«ey2/8ew2/8qK(x,w)qK(y, z1) dm(w) dm(y)Ã(â« Tâ2Kâsâsâï£®ï£°eâeK+u2 |z2âz1|2/4eK+u4 |z2 â z1|â§ 1ï£¹ï£» du)+CqT (x, z1)â«â« [(â« sâ0+â« Tâ2KTâ2Kâsâ)P Yy (|Yu| â¥ eK+u4 |z2 â z1|â£â£YTâ2K = w) du]Ã qK(x,w)qTâ2K(w, y)qK(y, z1) dm(w) dm(y)=: δ1 + δ2 + δ3 + δ4 + δ5, (2.6.18)where δi = δi(T,K, z1, z2) and is defined by the obvious correspondence. We first note thatδi(T,K, z1, z2)â 0 as K ââ (uniformly in T â¥ 2K) for i = 1, 2, 3.Turning to δ4 and δ5, we observe that by Lemma 2.2.2, eλ0T qT (x, z1)â Ï0(x)Ï0(z1) as Î»ââ. Since T â qT (x, z1) is continuous, qT (x, z1) > 0 for all T â¥ Ï(s) and Ï0(x)Ï0(z1) > 0,this implies that there exists β(x, z1) = β > 0 such thatqT (x, z1) â¥ Î²eâÎ»0TÏ0(x)Ï0(z1) â T â¥ Ï(s). (2.6.19)Applying (2.2.2) twice with δ = 1/8 and using (2.6.19), we have|δ4(T,K, z1, z2)|â¤ CÎ²â1Ï0(x)Ï0(z1)e2Î»0Keâ2Î»0Ke(x2+z21)/8â«â«ey2/4ew2/4dm(w) dm(y)Ã(â« Tâ2KâsâsâeâeK+u2 |z2âz1|2/4eK+u4 |z2 â z1|du)(2.6.20)â¤ Ce(x2+z21)/8Ï0(x)Ï0(z1)(â« Tâ2KâsâsâeâeK+u2 |z2âz1|2/4eK+u4 |z2 â z1|du)=4Ce(x2+z21)/8Ï0(x)Ï0(z1)â« âesâ/4eK/4|z2âz1|eâa2/4a2da, (2.6.21)where the last line follows from a simple substitution. Thus we have δ4(T,K, z1, z2)â 0 as109K â â, and again we note that convergence is uniform in T > 2K. It remains to handleδ5. By three applications of (2.2.2) with δ = 1/8 and (2.6.19), we have|δ5(T,K, z1, z2)|â¤ CÏ0(x)Ï0(z1)â«â« [(â« sâ0+â« Tâ2KTâ2Kâsâ)P Yy (|Yu| â¥ eK+u4 |z2 â z1|â£â£YTâ2K = w) du]Ã ex2/8ew2/4ey2/4ez21/8 dm(w) dm(y).The square bracketed term vanishes as K â â uniformly in T â¥ 2K + sâ(1/8) byLemma 2.6.1(b). The probabilities are bounded so the integrand obviously has a uniformlyintegrable upper bound. By Dominated Convergence, we have that δ5(T,K, z1, z2) â 0 asK ââ, uniformly in T â¥ 2K+sâ(1/8). We have therefore shown thatâ5i=1 δi(T,K, z1, z2)is arbitrarily small as K ââ, uniformly in T â¥ 2K + sâ(1/8) and in Î»â² > λ¯(s), where werecall that we have assumed λ, Î»â² > λ¯(s). From (2.5.33), λ > λ¯(s) is equivalent to T > Ï(s).As Ï(s) â¥ sâ(1/8), T â¥ 2K + Ï(s) implies that T â¥ 2K + sâ(1/8). Thus by (2.6.18) wehave proved the following. Recall that ZËT (Y, Î»â²/Î») = ZËT (Y, z1, z2, Î»â²/Î»).Lemma 2.6.2. For all x, z1, z2 â R such that z1 6= z2,Î´â¼a (K) = supTâ¥2K+Ï(s), Î»â²>Î»¯(s)â£â£EYx (ZËT (Y, Î»â²/Î»)â ZaT (Y, Î»â²/Î»,K) â£â£ YT = z1)â£â£satisfies limKââ Î´â¼a (K) = 0.Given this lemma, it suffices to find the limit of (the conditional expectation of)ZaT (Y, Î»â²/Î»,K), and so A2 has been replaced by 1.Next we consider A3(y, z1, λ, Î»â²,K), which we recall from (2.6.5) is defined asEYy(exp(â« K0F (Yu)âHÎ»â²/Î»eTâK+u(Yu, Yu + eKâu2 (z2 â z1)) du)â£â£â£â£YK = z1).We will show that in the limit as λ, Î»â² ââ, the integrand will be FâF2. Define Aâ3(y, z1,K)byAâ3(y, z1,K) = EYy(exp(â« K0F (Yu)âF2(Yu, Yu+eKâu2 (z2âz1)) du)â£â£â£â£YK = z1). (2.6.22)The difference between the integrands of A3 and Aâ3 is equal to (F2 â HÎ»â²/Î»eTâK+u)(Yu, Yu +eKâu2 (z2âz1)), which is non-negative by monotonicity. To obtain an upper bound we apply110Lemma 2.3.2 to obtain(F2 âHÎ»â²/Î»eTâK+u)(Yu, Yu + eKâu2 (z2 â z1))â¤ C[eâ(TâK+u)(2Î»0â1)/2 +(Î»â²Î»)â(2Î»0â1)eâ(TâK+u)(2Î»0â1)/2]â¤ Ce(Kâu)(2Î»0â1)/2(tâ s)â(2Î»0â1)/2[Î»â(2Î»0â1) + Î»â²â(2Î»0â1)]. (2.6.23)The first line uses the definition of Hc, which we recall from (2.5.12), and in the second linewe have used that T = log(Î»2(tâ s)). Since λ, Î»â² > λ¯(s) â¥ 1, the last expression in (2.6.23)is bounded by CeK/2(tâs)â(2Î»0â1)/2 for all u â [0,K]. Thus, using |exâey| â¤ (exâ¨ey)|xây|and (2.6.23), we have|Aâ3(y,z1,K)âA3(y, z1, λ, Î»â²,K)|â¤ exp(CKeK/2(tâ s)â(2Î»0â1)/2)Ã EYy(â« K0(F2 âHÎ»â²/Î»eTâK+u)(Yu, Yu + eK/2(z2 â z1)) duâ£â£â£â£YK = z1)â¤ exp(CKeK/2(tâ s)â(2Î»0â1)/2)(tâ s)â(2Î»0â1)/2Ã[Î»â(2Î»0â1) + Î»â²â(2Î»0â1)] â« K0Ce(Kâu)(2Î»0â1)/2duâ¤ C(K, tâ s)[Î»â(2Î»0â1) + Î»â²â(2Î»0â1)](2.6.24)for some constant C(K, t â s) > 0. Define ZbT (Y, Î»â²/Î»,K) as we defined ZaT (Y, Î»â²/Î»,K) in(2.6.7) but with F â F2 replacing the integrand in the second term. That is,ZbT (Y, Î»â²/Î»,K) = exp(â« K0F (Yu)âHÎ»â²/Î»eu (Yu, Yu + eTâu2 (z2 â z1)) du)Ã exp(â« TTâKF (Yu)â F2(Yu, Yu + eTâu2 (z2 â z1)) du). (2.6.25)In particular, we haveEYx(ZbT (Y, Î»â²/Î»,K)â£â£ YT = z1) (2.6.26)=â«â«A1(x,w, λ, Î»â²,K)Aâ3(y, z1,K)qK(x,w) qTâ2K(w, y) qK(y, z1)qT (x, z1)dm(w) dm(y).Because (2.6.24) is uniform in y and z1 and |A1| â¤ CZ , we can integrate out the transitiondensities to obtain the following.111Lemma 2.6.3. For K > 0 and s â [0, t), there is a constant C(K, tâ s) such thatδab (K,λ, Î»â²) =â£â£EYx (ZaT (Y, Î»â²/Î»,K) â ZbT (Y, Î»â²/Î»,K) â£â£YT = z1)â£â£â¤ C(K, tâ s)[Î»â(2Î»0â1) + Î»â²â(2Î»0â1)]for all λ, Î»â² > λ¯(s).We now analyse Aâ3 in greater detail. In particular, we perform a time reversal on theprocess Y . By (2.2.12), we haveAâ3(y, z1,K) = EYz1(exp(â« K0F (Yu)â F2(Yu, Yu + eu2 (z2 â z1)) du)â£â£â£â£YK = y).This is the term that in (2.6.1) we claimed converges to EY,âZ1 (Wâ(Y, z2)), defined in(2.5.39), in the limit. However, the above expectation is still conditional on the end-point. We now show that the contribution from the tail of the integral is vanishing, makingthe quantity asymptotically independent of the endpoint y. Let 0 < M < K. DefineAâ3(y, z1,M,K) by truncating the integral in (2.6.22) at time M . That is,Aâ3(y, z1,M,K) = EYz1(exp(â« M0F (Yu)âF2(Yu, Yu+eu2 (z2âz1)) du)â£â£â£â£YK = y). (2.6.27)We now define Zc(Y, Î»â²/Î»,M,K) by truncating the corresponding integral in Zb(Y, Î»â²/Î»,K)(the integral over [T â K,T ] in (2.6.25) becomes the integral over [T â M,T ]) so thatAâ3(y, z1,M,K) replaces Aâ3(y, z1,K) in the conditional expectation.Lemma 2.6.4. For all x, z1, z2 â R such that z1 6= z2,δbc(M) = supKâ¥M+sâ(1/8)supTâ¥2K+Ï(s), Î»â²>Î»¯(s)â£â£EYx (ZbT (Y, Î»â²/Î»,K)âZcT (Y, Î»â²/Î»,M,K) â£â£ YT = z1)â£â£satisfies limMââ δbc(M) = 0.Proof. Using the inequality |eâx â eây| â¤ |xâ y| for x, y â¥ 0, we have|Aâ3(y, z1,K)âAâ3(y, z1,M,K)|â¤ EYz1(â« KM|F (Yu)â F2(Yu, Yu + eu/2(z2 â z1))| duâ£â£â£â£ YK = y). (2.6.28)By Proposition 2.3.1(b), the absolute value of the above integrand is at most F (Yu +eu2 (z2â z1)) (for a similar argument see (2.6.11)). Exchanging expectation and integration,we proceed as in (2.6.14), (2.6.15), and (2.6.16), and apply Lemma 2.6.1 to bound (2.6.28)112above byc1â« KâM0(1 + eM+u4 |z2 â z1|)eâeM+u2 |z2âz1|2/2 du+ F (0)â« KMP Yz1(|Yu| â¥ eu4 |z2 â z1|â£â£YK = y) duâ¤ 4c1â« âeM/4|z2âz1|(1 + aâ1)eâa2/2 da+CeâÎ»0KqK(z1, y)e(z21+y2)/8â« KâMâsâ0ï£®ï£°eâeM+u2 |z2âz1|2eM+u4 |z2 â z1|â§ 1ï£¹ï£» du+ Câ« KKâsâP Yz1(|Yu| â¥ eM+u4 |z2 â z1|â£â£YK = y) du. (2.6.29)Expanding in terms of transition densities and using |A1| â¤ CZ , we haveâ£â£EYx (ZbT (Y, Î»â²/Î»,K) â ZcT (Y, Î»â²/Î»,M,K) â£â£ YT = z1)â£â£â¤ CqT (x, z1)â«|Aâ3(y, z1,K)âAâ3(y, z1,M,K)|qTâK(x, y) qK(y, z1) dm(y).Using (2.6.29) as an upper bound for the integrand, we obtain an expression which closelyresembles (2.6.18); in particular, four terms appear, directly corresponding to δ2, δ3, δ4 andδ5 of that expression. Moreover, they can be handled using the exact same arguments, asin the proof of Lemma 2.6.2, but with (M,K) playing the roles of (K,T ). Because thearguments are the same, we omit them.We now establish the limit of Aâ3(y, z1,M,K) as K â â. Recalling (2.6.27) and thedefinition of WM in (2.5.39), we haveAâ3(y, z1,M,K) = EYz1(exp(â« M0F (Yu)â F2(Yu, Yu + eu/2(z2 â z1)) du)â£â£â£â£YK = y)= EYz1(WM (Y, z2)â£â£YK = y).The functional WM (Y, z2) is a bounded continuous function of Y |[0,M ]. By Lemma 2.2.3(b),we haveâM > 0, limKââAâ3(y, z1,M,K) = EY,âz1 (WM (Y, z2)) . (2.6.30)113We define ZdT (Y, Î»â²/Î»,M,K) byZdT (Y, Î»â²/Î»,M,K):= exp(â« K0F (Yu)âHÎ»â²/Î»eK+u(Yu, Yu + eTâu2 (z2 â z1)) du)EY,âz1 (WM (Y, z2)) . (2.6.31)Note that the second term is now deterministic; it no longer depends on the originalOrnstein-Uhlenbeck process Y or the spatial variable y. We then haveEYx(ZdT (Y, Î»â²/Î»,M,K)â£â£ZT = z1)= EY,âz1 (WM (Y, z2))â«A1(x,w, λ, Î»â²,K)qK(x,w) qTâK(w, z1)qT (x, z1)dm(w).Bounding A1 â¤ CZ and integrating out the transition densities, by (2.6.30) we obtain thefollowing.Lemma 2.6.5. For all x, z1, z2 â R,δcd(M,K) = supTâ¥2K+Ï(s), Î»â²>Î»¯(s)â£â£EYx (ZcT (Y, Î»â²/Î»,M,K)â ZdT (Y, Î»â²/Î»,M,K) â£â£ YT = z1)â£â£satisfies limKââ δcd(M,K) = 0 for each fixed M > 0.From our starting expression for ZËT (Y, Î»â²/Î») in (2.6.6), all that remains to be handledin ZdT (Y, Î»â²/Î»,M,K) is the A1 term, whose definition we recall from (2.6.3) isA1(x,w, λ, Î»â²,K) = EYx(exp(â« K0F (Yu)âHÎ»â²/Î»eu (Yu, Yu + eTâu2 (z2 â z1)) du)â£â£â£â£YK = w).The dominant contribution to the integral in A1 resembles F (Yu)âV eu/21 (Yu). By Proposi-tion 2.3.1 we have the following upper and lower bounds for the difference of this quantityand the integrand:0 â¤[F (Yu)â V eu/21 (Yu)]â[F (Yu)âHÎ»â²/Î»eu (Yu, Yu + eTâu2 (z2 â z1))]â¤ F (Yu + eTâu2 (z2 â z1)). (2.6.32)Recall from (2.3.16) that ZK(Y ) is defined asZK(Y ) = exp(â« K0F (Yu)â V eu/21 (Yu) du).Because the exponentials in both A1 and ZK(Y ) are bounded above by CZ , by (2.6.32) we114haveâ£â£A1(x,w, λ, Î»â²,K)â EYx (ZK(Y ) |YK = w)â£â£â¤ CZ EYx(â« K0F (Yu + eTâu2 (z2 â z1)) duâ£â£â£â£ YK = w). (2.6.33)We define Ze(Y,M,K) byZe(Y,M,K) = ZK(Y )Ã EY,âz1 (WM (Y, z2)) . (2.6.34)Using (2.6.33) and proceeding as in the proofs of Lemmas 2.6.2 and 2.6.4, we obtain thefollowing.Lemma 2.6.6. For all x, z1, z2 â R such that z1 6= z2,δde (T,M,K) = supÎ»â²>Î»¯(s)â£â£EYx (ZdT (Y, Î»â²/Î»,M,K)â Ze(Y,M,K) â£â£YT = z1)â£â£satisfies limTââ δde (T,M,K) = 0 for all fixed M and K such that 0 < M < K.From (2.6.34), we haveEYx (Ze(Y,M,K) |YT = z1) = EYx (ZK(Y ) |YT = z1)EY,âz1 (WM (Y, z2)) .Thus by Lemma 2.2.3(b) and the fact that ZK(Y ) â¤ CZ (and is a continuous function ofY ) we have the following.Lemma 2.6.7. For all x, z1, z2 â R,δef (T,M,K) =â£â£EYx (Ze(Y,M,K) |YT = z1)â EY,âx (ZK(Y ))EY,âz1 (WM(Y, z2)) â£â£satisfies limTââ δef (T,M,K) = 0 for all fixed M and K such that 0 < M < K.We are now ready to establish the limiting form of EYx (ZËT (Y, Î»â²/Î») |YT = z1) (providedz1 6= z2). Let M > 0, K > M , T â¥ 2K + Ï(s) and Î»â² > λ¯(s). Bounding above by the sumof the δ terms in Lemmas 2.6.2-2.6.7, we have thatâ£â£EYx (ZËT (Y, Î»â²/Î») |YT = z1)â EY,âx (ZK(Y ))EY,âz1 (WM (Y, z2)) â£â£â¤ Î´â¼a (K) + δab (K,λ, Î»â²) + δbc(M) + δcd(M,K) + δde (T,M,K) + δef (T,M,K). (2.6.35)Let Ç« > 0. By Lemma 2.6.4, we can choose M0 > 0 to be sufficiently large such thatδbc(M) < Ç«/4 for allM â¥M0, and choose someM â¥M0. By Lemma 2.6.2 and Lemma 2.6.5,115we can then chooseK0 to be large enough such that Î´â¼a (K)+δcd(M,K) < Ç«/4 for allK â¥ K0.Fix K > K0. Next, by Lemmas 2.6.6 and 2.6.7 we can choose T0 > 2K + Ï(s) such thatfor all T â¥ T0, δde (T,M,K) + δef (T,M,K) < Ç«/4. Finally, Lemma 2.6.3 allows us to chooseÎ»(Ç«) > λ¯(s) such that T = log(λ2(t â s)) â¥ T0 and δab (K,λ, Î»â²) < Ç«/4 for all λ, Î»â² â¥ λ(ǫ).We therefore obtain from (2.6.35) thatlim supÎ»,Î»â²âââ£â£EYx (ZËT (Y, Î»â²/Î») |YT = z1)â EY,âx (ZK(Y ))EY,âz1 (WM (Y, z2)) â£â£ < ǫfor the M and K chosen above. This holds for all Ç« > 0 for sufficiently large M and K(with M < K). It therefore holds that if limM,Kââ,K>M EY,âx (ZK(Y ))EY,âz1 (WM (Y, z2))exists, then limÎ»,Î»â²ââEYx (ZËT (Y, Î»â²/Î») |YT = z1) exists and is equal to it. Thus it sufficesto find the limit of EY,âx (ZK(Y ))EY,âz1 (WM(Y, z2)) as M,K â â with M < K. As thefirst term depends only on K and the second depends only onM , we can consider the limitsindependently. First consider EY,âx (ZK(Y )). By (2.3.18), ZK â Zâ â¤ CZ , so the limit of thefirst term as K ââ is EY,âx (Zâ(Y )) by Monotone Convergence. We recall the definitionof WM from (2.5.39). The integral in WM is monotone in M and hence converges to theintegral on [0,â] as M ââ. Using the fact that |WM (Y, z2)| â¤ 1 for all M and continuityof the exponential, we can bring the limit inside, and EY,âz1 (WM (Y, z2))â EY,âz1 (Wâ(Y, z2))as M ââ. Thus we have shown thatlimÎ»,Î»â²ââEYx (ZËT (Y, Î»â²/Î») |YT = z1) = EY,âx (Zâ(Y ))EY,âz1 (Wâ(Y, z2)) .This is (2.6.1), which is what we wanted to show, so the proof of Lemma 2.5.5 is complete.116Chapter 3The boundary local time ofsuper-Brownian motion II: 0-1 lawSummary. If X(t, x) is the density of one-dimensional super-Brownian motion, we provethatdim(â{x : X(t, x) > 0}) = 2â 2λ0 â (0, 1) a.s. on {Xt 6= 0},where âÎ»0 â (â1,â1/2) is the lead eigenvalue of a killed Ornstein-Uhlenbeck process.This confirms a conjecture of Mueller, Mytnik and Perkins [69] who proved the above withpositive probability. To establish this result we derive some new basic properties of theboundary local time introduced in Chapter 2 analyse the behaviour of X(t, Â·) near theupper edge of its support. Numerical estimates of λ0 suggest that the above Hausdorffdimension is approximately .224.1173.1 Introduction and statement of resultsLet (Xt, t â¥ 0) denote a super-Brownian motion on the line starting at X0 6= 0 underPXX0 . Here X0 â MF (R), the space of finite measures on R with the topology of weakconvergence, and PXX0 will denote any probability under which X has the above law. Ourbranching rate is chosen to be one so that the jointly continuous density, X(t, x), of Xt fort > 0, is the unique in law solution of the stochastic partial differential equation (SPDE)âXât(t, x) =12â2Xâx2(t, x) +âX(t, x)WË (t, x), X â¥ 0, X(0) = X0 (3.1.1)(see Section III.4 of [83]). Here WË is a space-time white noise on [0,â)ÃR, and the initialcondition means that Xt(dx) = X(t, x)dxâ X0(dx) in MF (R) as t â 0.The boundary of the zero set of Xt,âZt = â{x : X(t, x) = 0} = â{x : X(t, x) > 0}, (3.1.2)was studied in [69]. The increased regularity of X on and near this set has played animportant role in the study of SPDE such as (3.1.1) (see [74] and [72]). Mytnik and Perkins(unpublished) had obtained side conditions on X which would give pathwise uniquenessin (3.1.1) but which would imply that dim(âZt), the Hausdorff dimension of âZt, is zero.The intuition here is that solutions to (3.1.1) should only separate in their respective zerosets since these are the only points at which the noise coefficient is non-Lipschitz. So thesmaller this set is, the harder it will be for solutions to separate. In [69] it was shownthat if âÎ»0 â (â1,â1/2) is the lead eigenvalue of the killed Ornstein-Uhlenbeck operatordescribed below, then (see Theorem 1.3 of [69]) in factPXX0(dim(âZt) = 2â 2λ0) > 0. (3.1.3)Here it was also conjectured (see the comment following Theorem 1.3 in [69]) thatdim(âZt) = 2â 2λ0 a.s. on {Xt 6= 0}. (3.1.4)In any case, the rigorous bounds on λ0 mentioned above imply the dimension of âZt isin (0, 1), at least with positive probability, and the aforementioned pathwise uniquenessproblem remains unresolved in spite of a recent negative result in Chen [9]. Here pathwisenon-uniqueness to (3.1.1) was shown if an innocent looking immigration term of the formÏ(x) (Ï smooth, non-negative and compactly supported) is added to the right-hand side of(3.1.1). The immigration term, however, gives âZt positive Lebesgue measure and this iswhat allows Chen to establish separation of solutions.118The boundary set itself is rather delicate as small perturbations of X will of coursecompletely change the nature of âZt. In particular, it is a non-monotone function of theinitial condition. This is one reason some of the standard zero-one arguments (see, e.g., theproof of Theorem 1.3 of [80] for the dimension of the range of X) were not able to resolvethe conjecture (3.1.4). Our main result (Theorem (3.1.1) below) will use the boundary localtime Lt(dx) of âZt, which was constructed in Chapter 2, to confirm (3.1.4). The local timeis a random measure supported by âZt which we are just beginning to understand, and someof its basic properties derived here will play a central role in our arguments. As a randommeasure supported on the set of points where solutions to (3.1.1) can separate, Lt(dx) hasthe potential of playing the same role in the study of SPDE arising from population modelsthat ordinary local time does for stochastic differential equations. Of course, one wouldneed to construct Lt for a much larger class of random processes.Numerical estimates of λ0 due to Peiyuan Zhu suggest that (3.1.4) impliesdim(âZt) â .224 a.s. on {Xt(1) > 0}, (3.1.5)perhaps larger than one may think given that X(t, Â·) is Ho¨lder 1â η in space near its zeroset for any η > 0 (see Theorem 2.3 in [72]). We briefly discuss this approximation belowand give some evidence for the accuracy of the estimate to the digits given.It will often be more convenient to work with the canonical measure of super-Brownianmotion, Nx, which is a more fundamental object in many ways. Recall that X arises as thescaling limit of the empirical measures of critical branching random walk. Nx is a Ï-finitemeasure on C([0,â),MF (R)) (the space of continuous measure-valued paths) describingthe behaviour of the descendants of a single ancestor at x at time 0 (see Theorem II.7.3 of[83]). A super-Brownian motion under PXX0 may be constructed as the integral of a Poissonpoint process with intensity NX0(Â·) =â«Nx(Â·)dX0(x) (see (3.2.5) below). In particular, ifwe write Xt(Ï) =â«Ï(x)Xt(dx), then for Ï â¥ 0,EXX0(eâXt(Ï)) = exp(ââ«1â eâÎ½t(Ï)dNX0(Î½)). (3.1.6)Our next job is to describe λ0 more carefully. We letF (x) = â log(PXδ0 (X(1, x) = 0)) = N0(X(1, x) > 0), (3.1.7)where the last equality is a simple consequence of (3.1.6) with Ï = âÎ´x and X0 = δ0 (seeProposition 3.3 of [69]). Then F is the unique positive symmetric C2 solution toF â²â²2(y) +y2F â²(y) + F (y)â F (y)22= 0, (3.1.8)119andF â²(0) = 0, limyââ y2F (y) = 0. (3.1.9)(See (1.10) and (1.12) of [69] and the discussion in Section 3 of the same reference.) LetAf(y) = fâ²â²(y)2 â yfâ²(y)2 be the generator of the Ornstein-Uhlenbeck process, Y , on the line.For Ï â C([ââ,â]), the space of continuous functions on R with finite limits at Â±â, we letAÏ(f) = Af â Ïf be the generator of the Ornstein-Uhlenbeck process Y Ï, now killed whenâ« t0 Ï(Ys) ds exceeds an independent exponential mean one r.v. If m denotes the standardnormal law on R, the resolvent of AÏ is a Hilbert-Schmidt integral operator on the Hilbertspace of square integrable functions with respect to m, L2(m). Therefore AÏ has a completeorthonormal system of eigenfunctions {ÏÏn : n â¥ 0} with non-positive eigenvalues {âÎ»Ïn}ordered so that âÎ»Ïn decreases to ââ. The lead eigenvalue âÎ»Ï0 â¤ 0 is simple and sohas a unique normalized eigenfunction ÏÏ0 . See Theorem 3.2.1 below for this and relatedinformation. If we set Ï = F , then our eigenvalue âÎ»0 is âÎ»F0 , which is in (â1,â1/2) byan elementary calculation in Proposition 3.4(b) of [69], using the fact (Proposition 3.4(b)of [69]) thatλF/20 = 1/2. (3.1.10)Here then is our main result.Theorem 3.1.1. For any X0 â MF (R) {0} and t > 0,dim(âZt) = 2â 2λ0 â (0, 1) PXX0 â a.s. and N0 â a.e. on {Xt > 0}. (3.1.11)In fact Theorem 1.3(a) in [69] already givesdim(âZt) â¤ 2â 2λ0 PXX0 â a.s. and N0 â a.e. (3.1.12)Although the above reference only considers PXX0 , the result for N0 then follows easily bythe Poisson point process decomposition mentioned above (see (2.5) below), just as in thelast six lines of the proof of Theorem 3.1.2 at the end of Section 3.4. Therefore it is thelower bound on dim(âZt) that we must consider. The lower bound on the dimension wasattained with positive probability in Theorem 5.5 of [69] by first deriving a sufficient capacitycondition for âZt to intersect a given set, A, with positive probability (Theorem 5.2 of [69])and then taking A to be the range of an appropriate Le´vy process. As was already noted,the authors were unable to use this approach to establish the lower bound a.s. The standardapproach to lower bounds on Hausdorff dimension is through the energy method. That is,120first construct a finite random measure or local time, Lt, supported by âZt such thatE(â«â«|xâ y|âÎ±dLt(x)dLt(y))<â â 0 < α < 2â 2λ0. (3.1.13)The energy method (see Theorem 4.27 of [65]) would then implydim(âZt) â¥ 2â 2λ0 a.s. on {Lt 6= 0}. (3.1.14)The existence of such a boundary local time was established in [35], confirming a con-struction conjectured in Section 5 of [69], which we briefly describe now. Define a measureLλt â MF (R) byLλt (Ï) =â«Ï(x)Î»2Î»0X(t, x)eâÎ»X(t,x) dx (3.1.15)for bounded Borel functions Ï. Note that as λ gets large Lλt becomes concentrated on theset of points x where 0 < X(t, x) = O(1/λ). The normalization of λ2λ0 comes from theleft tail behaviour of X(t, x) in Theorem 1.2 of [69]. The following result is taken fromChapter 2, more specifically it is included in Theorems 2.1.1, 2.1.2, 2.1.3 and 2.1.5, andProposition 2.1.6.Theorem 3.A. (a) There is a finite atomless random measure, Lt, on the line such thatunder N0 or PXX0,Lλt â Lt in measure in the metric space MF (R) as Î»ââ.Moreover Lt is supported on âZt a.s.(b) There is a positive constant CA such that for any Borel Ï : Râ [0,â),â«Lt(Ï)dN0 = CAtâÎ»0â«Ï(âtz)ÏF0 (z)dm(z). (3.1.16)(c) (3.1.13) holds under both N0 and PXX0.(d) There is a constant CB such thatâ«Lt(1)2dN0 â¤ CBt1â2Î»0 . (3.1.17)Let S(Xt) = {x : X(t, x) > 0} be the closed support of Xt and define Ut = sup(S(Xt)) tobe the upper most point of the support. It now follows from Theorem 3.A that (3.1.14) holdsunder both PXX0 and N0 (for the latter one can work under the probability N0(Â·|Xt 6= 0)).And so Theorem 3.1.1 is immediate from (3.1.12) and the following:121Theorem 3.1.2. Under the measures N0 and PXX0, Lt > 0 almost surely on {Xt > 0}. Infact, almost surely on {Xt > 0}, Lt((Ut â δ, Ut)) > 0 for all δ > 0.This theorem shows that as long as Xt has not gone extinct, the part of âZt at itsupper edge will have positive Lt measure, and, in particular, Lt itself is not equal to thezero measure. It is natural to consider a local version of the above and show that Ltwill charge any open interval which contains points in âZt. This clearly fails (note fromTheorem 3.A that Lt is atomless) if Xt(Â·) has isolated zeros, which clearly would be inâZt. An elementary argument shows that âS(Xt) â âZt and the former set clearly willnot contain isolated zeros of Xt(Â·). Given that the existence of isolated zeros of Xt(Â·)remains unresolved (we conjecture that they do not exist), here then is our local version ofTheorem 3.1.2:Theorem 3.1.3. For t > 0, PXX0-a.s. and N0-a.e., for any a < b, (a, b) â© âS(Xt) 6=â implies Lt((a, b)) > 0.Evidently we do not know whether or not âZt âS(Xt) is non-empty; isolated zeros arenot the only possible points in this setâsee Lemma 3.5.1 below. Nonetheless we make thefollowing conjecture:Conjecture. Lt is supported on âS(Xt) and so dim(âS(Xt)) = 2 â 2λ0 on {Xt 6= 0}PXX0-a.s. and N0-a.e.,the last conclusion being immediate from the first by (3.1.12), Theorem 3.1.2, Theo-rem 3.A(a,b) and the energy method described above.Corollary 3.1.4. For t > 0, PXX0-a.s. and N0-a.e., for any a < b, (a, b) â© âS(Xt) 6=â implies dim(âZt â© (a, b)) = 2â 2λ0.Proof. By considering rational values we may fix a and b and work under either PXX0 orN0(Â·|Xt 6= 0). Assume (a, b) â© âS(Xt) 6= â . In view of (3.1.13) we may apply the energymethod to Lt|(a,b), which is a.s. non-zero by Theorem 3.1.3, and so conclude that dim(âZtâ©(a, b)) â¥ 2 â 2λ0 a.s. on {(a, b) â© â(S(Xt)) 6= 0}. The corresponding upper bound isimmediate from (3.1.12).We comment briefly on the numerical approximation of λ0 carried out by Peiyuan Zhu in[96]. One first needs to numerically approximate F using an ODE solver and the âshootingmethodâ to find the minimal value of c so that Fc(0) = c, Fâ²c(0) = 0 and Fc satisfying (3.1.8)remains non-negative. It is known that Fc = F (see, e.g., [6]). One then approximates thisnumerically generated F by a linear combination of Gaussians FË (with varying meansand variances). We estimate âÎ»F0 by âÎ»FË0 , the lead eigenvalue of the Ornstein-Uhlenbeck122operator with FË -killing on a large interval [0,K] with Neumann boundary conditions. Kmust be taken sufficiently large to approximate the corresponding operator on [0,â). Thefinal step is then to use CHEBFUN software to estimate Î»FË0 . One could also obtain FË byinterpolating between the numerically generated grid points using Chebychev polynomialsâthe results agree to the given accuracy. We have some faith in the resulting approximationof λF0 â .8882 because if we replace F with F/2, the same method leads to λF/20 â .5000.This compares well with the exact (known) value in (3.1.10).The proof of Theorem 3.1.2 includes some input from the semilinear pdeâs associatedwith super-Brownian motion (such as (3.1.19) below) which are carried out in Section 3.3.This is then used in Section 3.4, to study Xt(dx) near the upper end of its support, Ut. Forǫ > 0, defineÏ Ç« = Ï Ç«(t) = inf{x â R : Xt([x,â)) < Ç«}. (3.1.18)In particular, if Xt(1) < Ç«, then ÏÇ« = ââ. The following result gives some insight intothe behaviour of Xt near the upper edge of its support and so the following first momentbound, which is proved in Section 3.4, may be of independent interest.Proposition 3.1.5. There is a non-increasing function, c3.1.5(t), such that for all t, Ç« > 0and u > 0:(a) For any X0 â MF (R), EXX0(â«âÏÇ«(t)âuX(t, x)dx)â¤ c3.1.5(t)X0(1)(u2 â¨ Ç«).(b) N0(â«âÏÇ«(t)âuXt(dx))â¤ c3.1.5(t)(u2 â¨ Ç«).One can understand the important u2 behaviour in the above for small u, Ç« from theimproved modulus of continuity of X(t, Â·) near its zero set (mentioned above). Theorem 2.3of [72] shows that for η > 0 there is Î´(Ï) > 0 so that |X(t, x) âX(t, x + h)| â¤ |h|1âÎ· forX(t, x) â¤ |h| â¤ Î´(Ï). This readily leads to (for Ç«, u small) X(t, Ï Ç«(t)) â¤ Ç«.5âÎ· and after ashort argument (consider u â¥ Ç«.5âÎ· and u < Ç«.5âÎ· separately) thatâ« UtÏÇ«(t)âuX(t, x)dx =â« ÏÇ«ÏÇ«(t)âuX(t, x) dx + Ç« â¤ c(Ç«1â2Î· + u2âÎ·),which comes close to the above mean behaviour. The actual proof uses the unique non-negative solution, vât (x) = vâ(t, x), in C1,2((0,â) Ã R) ofâvâât(t, x) =12â2vââx2â (vâ)22, vâ0 =â1(ââ,0]. (3.1.19)Such semilinear parabolic equations arise of course as exponential dual functions for super-Brownian motionâsee Section 3.3 for more on this in general, and Theorem 3.3.3 for moreinformation on the particular equation above, including its precise meaning. More specifi-cally, the proof uses G(x) = vâ1 (x) which also is the unique Câ solution of (3.1.8) but now123with the boundary conditions (see Lemma 3.3.4(c))limxââx2G(x) = 0, limxâââG(x) = 2. (3.1.20)Using a Palm measure formula for Xt (Theorem 4.1.3 from [15]), the Feynman-Kac Formulaand some pde bounds (notably Proposition 3.3.7), we show (see (3.4.10)) that for u2 â¥ Ç«(from which the general case follows easily),N0(â« âÏÇ«(t)âuX(t, x) dx)â¤ c(t)EYm(exp(ââ« log(1/u2)0G(Ys) ds)), (3.1.21)where Y is an (unkilled) Ornstein-Uhlenbeck process with initial law m under P Ym . So, as in[69], one can use the spectral decomposition of AG to see that the right-hand side of (3.1.21)is at most c(t)eâÎ»G0 log(1/u2) = c(t)u2λG0 . Unlike λF0 , we can identify the eigenfunction forλG0 and verify that λG0 = 1 (Proposition 3.3.5), and hence obtain the required bound inProposition 3.1.5(b).Turning to Theorem 3.1.2 itself, Theorem 3.A(b),(d) and the second moment methodeasily give (Lemma 3.4.1)N0(LÎ´([3âÎ´,â)) > 0|Xδ 6= 0) â¥ p > 0 â δ > 0.One can then use this to conclude that the right-most ancestor, say at x, at time tâÎ´ of thepopulation at time t will have descendants at time t with a positive boundary local time on[x+3âÎ´,â) with conditional (on FtâÎ´) probability at least p. Now one must show that thedescendants of the other ancestors at time t â δ do not flood into the boundary region ofthe right-most ancestor and hence remove it from the overall boundary. This issue capturesthe delicate and non-monotone character of the boundary. To resolve it we use a classicalhitting estimate for X from [14] (see Theorem 3.2.3 below) and Proposition 3.1.5. Thiswill lead to a uniform lower bound on PXX0(Lt > 0|FtâÎ´n) with high probability at least on{Xt 6= 0} and the martingale convergence theorem then shows Lt > 0 with high probabilityon {Xt 6= 0}.Theorem 3.1.3 is proved in Section 3.5. Section 3.2 reviews a number of standard tools wewill need in the proofs including the spectral decomposition of the killed Ornstein-Uhlenbeckprocesses, some cluster decompositions of super-Brownian motion based on historical infor-mation, and the aforementioned hitting estimate for super-Brownian motion.1243.2 Preliminaries3.2.1 Killed Ornstein-Uhlenbeck processesRecall that Y is an Ornstein-Uhlenbeck process with generator A, starting at x under P Yxand starting with the standard normal law m under P Ym . As above for Ï â C([ââ,â]),Ï â¥ 0, AÏ is the generator of the Ornstein-Uhlenbeck process, Y Ï, killed at time ÏÏ =inf{t : â« t0 Ï(Ys)ds > e}, where e denotes an independent exponential r.v. with mean one.The result below is standard, and included in Theorem 2.3 of Mueller, Mytnik and Perkins[69].Theorem 3.2.1. (a) AÏ has a complete orthonormal family {Ïn : n â¥ 1} of C2 eigenfunc-tions of L2(m) satisfying AÏÏn = âÎ»nÏn, where {âÎ»n}ân=1 is a non-increasing sequence ofnon-positive eigenvalues such that λn â â. Furthermore, âÎ»0 is a simple eigenvalue andÏ0 > 0.(b) Let θ =â«Ï0 dm. For all 0 < δ, there exists cδ such that for all x â R,|eλ0tP Yx (ÏÏ > t)â Î¸Ï0(x)| â¤ cÎ´eÎ´x2eâ(Î»1âÎ»0)t, (3.2.1)andÏ0(x) â¤ cδeδx2 . (3.2.2)In particular,P Yx (ÏÏ > t) â¤ CÎ´eÎ´x2eâÎ»0t. (3.2.3)andP Ym (ÏÏ > t) â¤ CeâÎ»0t. (3.2.4)3.2.2 Cluster and historical decompositions of super-Brownian motionWe recall the cluster decomposition of super-Brownian motion from Theorem 4 in Sec-tion IV.3 of [57]. If X0 â MF (R), let ÎX0 be a Poisson point process on the spaceC([0,â),MF (R)) of continuous measure-valued paths with intensity NX0(Â·) =â«Nx(Â·)dX0(x). ThenXt(Â·) =ï£±ï£²ï£³â«Î½t(Â·)ÎX0(dÎ½) if t > 0X0(Â·) if t = 0(3.2.5)defines a super-Brownian motion with initial state X0. In particular this shows that fort > 0, (D= denotes equality in law)XtD=Nâi=1Xit , (3.2.6)125where N has a Poisson law with mean 2X0(1)/t = NX0(Xt > 0), and given N , {Xit : i â¤ N}are iid random measures with law NX0(Xt â Â·|Xt > 0). The summands in (3.2.6) correspondto the contributions to Xt from each of the finite number of ancestors at time 0 of thepopulation at time t.We will also make use of the historical process associated with a super-Brownian mo-tion. The historical process encodes the genealogical information of the super-Brownianmotion X. Good introductions may be found in [15], or Sections II.8 and III.1 of [83].Let C([0,â),R) denote the space of continuous R-valued paths on [0,â), endowed withthe compact-open topology. The historical process (Ht : t â¥ 0) is a measure-valued time-inhomogeneous Markov process taking values in MF (C([0,â),R)) such that y· = ytâ§Â·for Ht-a.a. y for all t â¥ 0 a.s. If we identify constant paths with R, then, viewingH0 as an element of MF (R), we can recover the super-Brownian motion X starting atX0 = H0 from its associated historical process H by projecting Ht onto time t, that is,Xt(Â·) = Ht({y â C([0,â),R) : yt â Â·}). Intuitively, (ys, s â¤ t) gives the historical path ofthe particle yt in the support of Xt. We will use a modulus of continuity for the paths ygoverned by Ht. Let S(Ht) denote the closed support of Ht and set h(r) = (r log(1/r))1/2.For c > 0 and δ > 0, define K(c, δ) byK(c, δ) = {y â C([0,â),R) : |yr â ys| â¤ ch(r â s) â r, s â¥ 0 s.t. |r â s| â¤ δ}. (3.2.7)By Theorem III.1.3(a) of [83], if c > 2 and T > 0, then PXX0 -a.a. Ï, there exists δ =δ(T, c, Ï) > 0 a.s. such thatS(Ht) â K(c, δ) for all t â [0, T ]. (3.2.8)Moreover, the proof of the above shows that for any c > 2, T > 0 there are Ï(c) > 0 andC(T ) such thatPXX0(δ(T, c) â¤ r) â¤ C3.2.9(T )rÏ(c) for all r â (0, 1], (3.2.9)wherelimcââ Ï(c) =â. (3.2.10)A second decomposition of a superprocess based on historical information will also playan important role in our arguments. Let (Ft) be the usual right-continuous completedfiltration generated by H and assume 0 â¤ δ â¤ t are fixed. Assume Ï â [ââ,â] is aÏ(XtâÎ´)-measurable random variable. We decompose XtâÎ´ into the sum of two random126measures:XRtâÎ´(dx) = XR,Ï,Î´tâÎ´ (dx) = 1{xâ¥Ï}XtâÎ´(dx) and XLtâÎ´(dx) = XL,Ï,Î´tâÎ´ (dx) = 1{x<Ï}XtâÎ´(dx).(3.2.11)We then track the descendants of each of these populations at future times and so definemeasure-valued processes byXËRs (Ï) = XËR,Ï,Î´s (Ï) =â«Ï(ytâÎ´+s)1(ytâÎ´ â¥ Ï)HtâÎ´+s(dy)XËLs (Ï) = XËL,Ï,Î´s (Ï) =â«Ï(ytâÎ´+s)1(ytâÎ´ < Ï)HtâÎ´+s(dy).Clearly we haveXËRs + XËLs = XtâÎ´+s for all s â¥ 0, and (if s = 0) XRtâÎ´ +XLtâÎ´ = XtâÎ´ . (3.2.12)By (III.1.3) on p. 193 of [83] and the Markov property of H, we get:Conditional on FtâÎ´ , (XËRs ) and (XËLs ) are independent (FtâÎ´+s)-super-Brownian (3.2.13)motions with initial laws XRtâÎ´ and XLtâÎ´ , respectively.Given the above decompositions of super-Brownian motion into a sum of independentsuper-Brownian motions, it is not surprising that we will also need to know how the cor-responding boundary local time, Lt, decomposes. Recall that a sum of n independentsuper-Brownian motions with initial conditions X10 , . . . ,Xn0 is a super-Brownian motionstarting at X10 + · · · +Xn0 . The next result is Theorem 1.9 of [35].Theorem 3.2.2. Suppose X1, . . . ,Xn are independent one-dimensional super-Brownianmotions, starting at X10 , . . . ,Xn0 â MF (R) and with boundary local times L1t , . . . Lnt . LetX =âni=1Xi and Lt be the boundary local time of X. ThendLt(x) =nâi=11(âj 6=iXj(t, x) = 0)dLit(x) =nâi=11(X(t, x) = 0)dLit(x). (3.2.14)3.2.3 Hitting probabilities of super-Brownian motionThe proofs of our main theorems will make use of bounds on hitting probabilities for super-Brownian motion.Theorem 3.2.3. There exists a universal constant c3.2.3 <â such that:127(i) For R > 2ât,N0(Xs([R,â)) > 0 for some s â¤ t) â¤ c3.2.3Râ2(Rât)3eâR2/2t.(ii) For all X0 â MF (R) such that X0 is supported on (ââ, 0] and for all R > 2ât, wehavePXX0(Xs([R,â)) = 0 for all s â¤ t)â¥ exp(âc3.2.3â« 0ââ(Râ x)â2(Râ xât)3eâ(Râx)2/2tdX0(x)).Proof. (i) is a simple consequence of Theorem 3.3(b) of [14] with d = 1 (and its proof) and(3.2.5).We derive (ii) as a consequence of (i) by using (3.2.5). Indeed, this result and well-knownformulas for the Laplace transform of a Poisson point process (see, for example, Theorem24.14 of [43]) imply that for R > 2ât and θ > 0, we haveEXX0(exp(âÎ¸â« t0Xs([R,â)) ds))= exp(ââ«Nx(1â exp(âÎ¸â« t0Xs([R,â)) ds))dX0(x)).A simple application of Dominated Convergence allows us to let θ ââ and conclude thatPXX0(Xs([R,â)) = 0 for all s < t)= exp(ââ«Nx(Xs([R,â)) > 0 for some s â¤ t)dX0(x)).Part (ii) follows by applying (i) and translation invariance.3.3 Some semi-linear partial differential equationsWe recall the relationship of the Laplace functional of super-Brownian motion with solutionsof a semi-linear partial differential equation (PDE). We first present the integral form ofthe equation. Let Bb+(R) denote the space of non-negative bounded Borel functions on theline. Let EBx denote the expectation of standard Brownian motion with B0 = x, and denotethe Brownian semigroup by St, i.e. StÏ(x) = EBx (Ï(Bt)). By Theorem II.5.11 of [83], for128Ï â Bb+(R) there exists a unique non-negative solution to the integral equationvt = StÏââ« t0Stâs(v2s/2) ds for (t, x) â [0,â)Ã R, (3.3.1)which we denote by V Ït (x), such that for all X0 â MF (R),EXX0(eâXt(Ï))= eâX0(VÏt ). (3.3.2)It follows from (3.2.5) and the above with X0 = δx thatNx(1â eâXt(Ï)) = V Ït (x). (3.3.3)It is clear from (3.3.1) that V Ït (x) â¤ StÏ(x) â¤ âÏââ, and so V Ït (x) â StÏ(x) â 0 as t â 0pointwise in x. This readily implies thatV Ïtvâ Ï = V Ï0 as t â 0,wherevâ denotes vague convergence of the Radon measure V Ït (x)dx to Ï(x)dx. (3.3.1) isknown as the mild form of the PDEâvtât=12â2vtâx2â v2t2for (t, x) â (0,â) Ã R, vt vâ Ï = v0 as t â 0, (3.3.4)where it will be understood that solutions of (3.3.4) will be in the space C1,2((0,â) Ã R)of functions with continuous partial derivatives up to order 1 in time and 2 in space onthe given open set. This formulation allows one to consider initial conditions which aremeasures. In this context Marcus and Ve´ron [64] (Theorem 3.5) proved existence anduniqueness of a (non-negative) solution, V¯ Ï, to (3.3.4) as a rather special case of moregeneral initial conditions which they classify with their initial trace theory. The use of theirgeneral theory may seem like overkill, but it will soon be convenient to use a stability resultin [64]. It is easy to show that their solutions also satisfy the mild form (3.3.1) as we nowsketch. First, monotonicity of V¯ Ï in Ï (e.g. Theorem 3.4 of [64]) and comparison with theelementary solution with initial (constant) value âÏââ show thatV¯ Ï(t, x) â¤ âÏââ. (3.3.5)For ε > 0, V¯ Îµ,Ït := VÂ¯Ït+Îµ defines the unique solution to (3.3.4) with C2 initial data V¯ ÏÎµ andevidently the solution is now in C1,2([0,â) Ã R). Such strong solutions are known to besolutions of the mild equation (3.3.1) (see, e.g., the outline following Proposition II.5.10 in129[83] and use the above boundedness). We therefore haveV¯ Ït+Ç« = StVÂ¯ÏÇ« ââ« t0Stâs(VÂ¯Ï 2s+Ç« /2) ds for (t, x) â [0,â) ÃR.It is easy to justify taking the limit pointwise as Ç« â 0 (use (3.3.5)), which shows that V¯ Ïtsolves the integral equation (3.3.1). By uniqueness of solutions to (3.3.1) we conclude thatV¯ Ït = VÏt . We therefore have that for Ï â Bb+(R), there exists a unique non-negativesolution V Ït to (3.3.4) (also satisfying (3.3.1)) such that (3.3.2) and (3.3.3) hold.For λ > 0, we denote by vλt the unique non-negative solution ofâvtât=12â2vtâx2â v2t2, for (t, x) â (0,â) Ã R, vt vâ Î»1(ââ,0] = vλ0 as t â 0. (3.3.6)Given the above discussion, vλt is also a solution of (3.3.1) with Ï = Î»1(ââ,0]. We willsometimes write vλt (x) = vλ(t, x). By (3.3.3), translation invariance and symmetry, vλtsatisfies for all t > 0,vλt (x) = N0(1â eâÎ»Xt((ââ,âx])) = N0(1â eâÎ»Xt([x,â))). (3.3.7)Similarly, by (3.3.2) we also have for t > 0,EXÎ´0(eâÎ»Xt([x,â)))= eâvÎ»t (x). (3.3.8)It is an exercise to use uniqueness in (3.3.6) or scaling properties of super-Brownian motionto show that vλ satisfies the following scaling relationship:vλ(t, x) = rvÎ»/r(rt,ârx) â r, λ > 0. (3.3.9)Take r = λ to see thatvλ(t, x) = Î»v1(Î»t,âÎ»x) (3.3.10)and r = 1/t to obtainvλ(t, x) = tâ1vÎ»t(1, tâ1/2x). (3.3.11)The following monotonicity properties are clear from (3.3.7).Lemma 3.3.1. The map xâ vλt (x) is decreasing in x, and Î»â vλt (x) is increasing in λ.We may let λ â â in (3.3.7) and (3.3.8) and use Dominated Convergence (for this in(3.3.7), use 1â eâÎ»Xt([x,â)) â¤ 1(Xt([x,â)) > 0), which is integrable with respect to N0) to130see that vλ(t, x) â vâ(t, x) as λ â â, wherevât (x) = N0(Xt([x,â)) > 0)= â log (PXδ0 (Xt([x,â)) = 0) ). (3.3.12)Note thatvât (x) = N0(Xt([x,â)) > 0) â¤ N0(Xt(1) > 0) = 2/t, (3.3.13)(see Theorem II.7.2(iii) of [83]) and in particular vât is finite for t > 0.Proposition 3.3.2. vλt (x) â vât (x) uniformly on compact sets in (0,â) Ã R. In fact,there is uniform convergence for (t, x) â [a,â)Ã [âR,R], for any a > 0 and R > 0.Proof. Taking Î»ââ in (3.3.11), we obtain thatvât (x) = tâ1vâ1 (tâ1/2x). (3.3.14)This fact and (3.3.11) imply thatvât (x)â vλt (x) = tâ1[vâ1 (tâ1/2x)â vλt1 (tâ1/2x)].Let 0 < a,R. Then by the above, for t â¥ a we havevât (x)â vλt (x) â¤ aâ1[vâ1 (tâ1/2x)â vλa1 (tâ1/2x)],where we have used monotonicity in λ. Thussuptâ¥asup|x|â¤Rvât (x)â vλt (x) â¤ aâ1 suptâ¥asup|x|â¤Rvâ1 (tâ1/2x)â vλa1 (tâ1/2x)â¤ aâ1 sup|x|â¤aâ1/2Rvâ1 (x)â vλa1 (x).The continuity of vâ1 (e.g., from (3.3.12)) and Diniâs Theorem imply that vλ1 â vâ1 uniformlyon compact sets, and the result follows.Theorem 3.3.3. vât (x) â C1,2((0,â) Ã R) and is the unique non-negative solution to thePDE(i)âvtât=12â2vtâx2â v2t2on (t, x) â (0,â) Ã R,(ii) limtâ0â«Uvât (x) dx = +â âU â R open such that U â© (ââ, 0] 6= â , (3.3.15)limtâ0â«Kvât (x) dx = 0 âK â R compact such that K â (0,â).131Proof. From Proposition 3.3.2 we have the local uniform convergence of vλt to vât . Thefamily {vλt } therefore satisfies the conditions of Theorem 3.10 of [64], which shows thatvâ(t, x) solves (3.3.15). Uniqueness follows by Theorem 3.5 of the same paper.Recall (see (3.1.20)) that G(x) = vâ1 (x).Lemma 3.3.4. (a) For all t > 0, vât (x) = tâ1G(tâ1/2x) for all x â R.(b) G > 0 and is decreasing.(c) G â Câ(R) and is the unique positive C2 solution to the ordinary differential equation12Gâ²â²(x) +x2Gâ²(x) +G(x) â 12G(x)2 = 0 (3.3.16)with boundary conditions limxââ x2G(x) = 0, limxâââG(x) = 2.(d) There is a constant c3.3.4 such that:(i)G(x) â¤ c3.3.4|x|eâx2/2 âx > 2,0 â¤ 2âG(x) â¤ c3.3.4|x|eâx2/2 âx < â2.(ii) Gâ² â¤ 0, and|Gâ²(x)| â¤ c3.3.4x2eâx2/2 âx > 2|Gâ²(x)| â¤ c3.3.4eâx2/2 âx â¤ 0.Proof. (a) is a restatement of (3.3.14) and (b) is obvious from (3.3.12).We now prove (c). By Theorem 3.3.3, G = vâ1 is C2. Noting that vât (x) = tâ1G(tâ1/2x)solves (3.3.15)(i), we can use the chain rule to seeâ 1t2G(tâ1/2x)â 12t2(tâ1/2x)Gâ²(tâ1/2x) =12t2Gâ²â²(tâ1/2x)â 12t2G(tâ1/2x)2âG(y)â y2Gâ²(y) =12Gâ²â²(y)â 12G(y)2,This proves that G solves (3.3.16). To see that it is C3, we note that when we solve (3.3.16)for Gâ²â², the expression is differentiable because G and Gâ² are differentiable. Proceeding byinduction we see that G is Câ. The boundary conditions will clearly follow from (d) below.It remains to prove uniqueness. Let H be any positive C2 solution of (3.3.16) satisfying thegiven boundary conditions and set u(t, x) = tâ1H(tâ1/2x). Then, reversing the above stepsone easily sees that u is a C2 solution of (3.3.15)(i). Let 0 < a < b and choose t > 0 small132enough so that y2H(y) < ǫa for y â¥ atâ1/2. Thenâ« bav(t, x)dx = tâ1/2â« baH(tâ1/2x)tâ1/2dx â¤ tâ1/2â« âatâ1/2H(y)dyâ¤ Ç«atâ1/2â« âatâ1/2yâ2dy= Ç«.This proves the second boundary condition in (3.3.15)(ii). The first boundary conditionis even easier to establish. So by the uniqueness in Theorem 3.3.3, H(x) = u(1, x) =vâ(1, x) = G(x).Next, we prove part (d)(i). To deduce the bound for positive x, we note thatG(x) = N0 (X1([x,â)) > 0)â¤ N0 (Xs([x,â)) > 0 for some s â¤ 1)â¤ c3.2.3|x|eâx2/2,for all x > 2, by Theorem 3.2.3(i). The lower bound on 2âG(x) is immediate from (3.3.13)(for all x). For x < â2, we have2âG(x) = N0 (X1(1) > 0)â N0 (X1([x,â)) > 0)â¤ N0 (X1((ââ, x]) > 0) (3.3.17)â¤ c3.2.3|x|eâx2/2,again using Theorem 3.2.3(i) and symmetry.Finally consider part (d)(ii). By (b) Gâ² â¤ 0. Now note that (3.3.16) can be rewritten as(ex2/2Gâ²(x))â²= ex2/2G(x)(G(x) â 2).Integrating the above, for x0, x â R we getGâ²(x) = eâx2/2[ex20/2Gâ²(x0) +â« xx0ey2/2G(y)(G(y) â 2) dy]. (3.3.18)For x > x0 â¥ 2, both terms in the above are non-positive, and, if c is the provisional133constant arising in (i), we can use part (i) to deduce that|Gâ²(x)| â¤ eâx2/2[ex20/2|Gâ²(x0)|+ 2câ« xx0|y| dy]â¤ (c1(x0) + c2x2)eâx2/2 â¤ câ²x2ex2/2for x > 2. For x â¤ 0 = x0, we note that the integral in (3.3.18) has its sign reversed, sois positive. Because Gâ²(x) â¤ 0, |Gâ²(x)| is bounded above by the absolute value of the firstterm in (3.3.18), which gives the required bound.Recall from Section 3.2.1 that if G is as above, then AG is the generator of a killedOrnstein-Uhlenbeck process with killing function G, âÎ»G0 is its lead eigenvalue, and ÏG0denotes its corresponding unit eigenfunction in L2(m).Proposition 3.3.5. For some constant c3.3.5 > 0, ÏG0 (x) = âc3.3.5ex2/2Gâ²(x) with corre-sponding eigenvalue âÎ»G0 = â1.Proof. Recall that G is the Câ solution of (3.3.16). Rearranging the equation, we can writeGâ²â²(x) = âxGâ²(x)â 2G(x) +G(x)2. G is Câ, so we can differentiate again to obtain a newODE:12Gâ²â²â² +12Gâ² +12xGâ²â² +Gâ² â 122GGâ² = 0ââ 12Gâ²â²â² +12xGâ²â² +32Gâ² âGGâ² = 0 (3.3.19)Let Ï(x) = ex2/2Gâ²(x). Let us first observe that Ï â L2(m) becauseâ«Ï(x)2 dm(x) = (2Ï)â1/2â«Gâ²(x)2ex2/2 dx <â,where the integral converges by Lemma 3.3.4(d)(ii). We compute the first and secondderivatives of Ï:Ïâ²(x) = xex2/2Gâ²(x) + ex2/2Gâ²â²(x)andÏâ²â²(x) = x2ex2/2Gâ²(x) + ex2/2Gâ²(x) + xex2/2Gâ²â²(x) + xex2/2Gâ²â²(x) + ex2/2Gâ²â²â²(x)= x2ex2/2Gâ²(x) + ex2/2Gâ²(x) + 2xex2/2Gâ²â²(x) + ex2/2Gâ²â²â²(x).134Using the above, we evaluate AGÏ.AGÏ =12Ïâ²â²(x)â 12xÏâ²(x)â Ï(x)G(x)= ex2/2[12x2Gâ²(x) +12Gâ²(x) + xGâ²â²(x) +12Gâ²â²â²(x)]â ex2/2[12x2Gâ²(x) +12xGâ²â²(x)]â ex2/2[G(x)Gâ²(x)]= ex2/2[12Gâ²â²â²(x) +12xGâ²â²(x) +12Gâ²(x)âG(x)Gâ²(x)]= ex2/2[12Gâ²â²â²(x) +12xGâ²â²(x) +32Gâ²(x)âG(x)Gâ²(x)]â ex2/2Gâ²(x)= âÏ(x),where the last equality is due to (3.3.19). Moreover, Gâ²(x) â¤ 0 for all x, so âex2/2Gâ²(x) â¥ 0,and we have already seen that it is in L2(m). Therefore Ï is a non-positive eigenfunction ofAG with eigenvalue â1. Clearly Ï cannot be orthogonal to the lead eigenfunction ÏG0 > 0(recall Theorem 3.2.1(a)). It follows that ÏG0 = âc3.3.5Ï for some normalizing constantc3.3.5 > 0 and hence the corresponding lead eigenvalue is â1.The next result gives a bound on the left tail of the distribution of Xt([x,â)) whichwill play an important role in the proof of Proposition 3.1.5. We do not know what theâcorrectâ power law behaviour is, but see the remark at the end of this section for a possibleanswer.Proposition 3.3.6. For 0 < p < 1/6 and t > 0 there is a constant C3.3.6 = C3.3.6(p, t)such thatPXδ0(0 < Xt([x,â)) â¤ 1Î»)â¤ C3.3.6Î»âp for all x â R and λ > 0.Proof. It clearly suffices to consider λ â¥ 1, which is assumed until otherwise indicated. Let0 < p < 1/6 and ε = ε(p) â (0, 1/6â p). Assume we are working under a probability, P , forwhich H is a historical process defining the super-Brownian motion X starting at δ0 and letFt be the right-continuous completed filtration generated by H. E will denote expectationwith respect to P . Recall h(r), Ï(c) and δ(t, c) are as in (3.2.8) and (3.2.9). By (3.2.10) wemay choose c = c(p) large enough so that Ï(c)Îµ â¥ p and so by (3.2.9),P (δ(t + 1, c) â¤ Î»âÎµ) â¤ C3.2.9(t+ 1)Î»âÎµÏ(c) â¤ C3.2.9(t+ 1)Î»âp for all λ â¥ 1. (3.3.20)135By (II.5.11) and (II.5.12) of [83],EXX0(eâÎ»Xt(1)) = exp(â2Î»X0(1)2 + λt); PXX0(Xt(1) = 0) = exp(â2X0(1)/t), (3.3.21)and so for λ â¥ 1,P (0 < Xt(1) â¤ Î»â1/6) â¤ eE(1(Xt(1) > 0) exp(âÎ»1/6Xt(1)))= e[exp( â2Î»1/62 + Î»1/6t)â exp(â2/t)]â¤ 2e[1tâ λ1/62 + Î»1/6t]â¤ 4etâ2Î»â1/6. (3.3.22)LetE = Ex,λ = {0 < Xt([x,â)) â¤ 1/λ, δ(t+ 1, c) > Î»âÎµ,Xt(1) > Î»â1/6}.Then by (3.3.20) and (3.3.22), it suffices to showP (Ex,λ) â¤ C(t, p)Î»âp for all x â R and λ â¥ 1. (3.3.23)Assume for now that x â R and λ â¥ 1. Note that if Ï(Î») = ÏÎ»â2/3(t) (recall Ï ε is as in(3.1.18)), thenon Ex,λ we have, ââ < Ï(Î») < x and Xt([Ï(Î»),â)) = Î»â2/3. (3.3.24)IntroduceE1x,λ = E â© {xâ Ï(Î») â¥ Î»â1/6} and E2x,λ = E â© {xâ Ï(Î») < Î»â1/6}.We consider E1 first. Setβ =13+ ε.Then there is a λ = λ(c, ε, t) = λ(p, t) â¥ 1 such that2ch(Î»âÎ²) < Î»â1/6 and Î»âÎ² < t/2 for λ â¥ λ. (3.3.25)Until otherwise indicated we will assume now that λ â¥ λ. DefineÎ¶tâÎ»âÎ² = inf{s â¥ 0 : Hs+tâÎ»âÎ² ({y : ytâÎ»âÎ² â¥ xâ ch(Î»âÎ²)}) = 0}.136It follows from (3.2.13) that for u = 0 or Î»âÎ²,conditional on Ftâu, Zs = Htâu+s({y : ytâÎ»âÎ² â¥ xâ ch(Î»âÎ²)}) (s â¥ 0) is equal in law tothe Feller diffusion (Xs(1), s â¥ 0) starting at Htâu({y : ytâÎ»âÎ² â¥ xâ ch(Î»âÎ²)}). (3.3.26)Throughout this proof we will assume the Feller diffusion Xs(1) starts at x0 â¥ 0 under Px0 .On E1 we have Î»âÎ² â¤ Î»âÎµ < δ(t+ 1, c) and so by the modulus of continuity (3.2.8),Ht({y : ytâÎ»âÎ² â¥ xâ ch(Î»âÎ²)}) â¥ Ht({y : yt â¥ x}) = Xt([x,â)) > 0.This implies that (use (3.3.26) with u = Î»âÎ² to see that Zs sticks at zero when it hits zero)Î¶tâÎ»âÎ² > Î»âÎ² on E1x,λ. (3.3.27)Now again use the modulus of continuity and then (3.3.25) to that on E1,Ht({y : ytâÎ»âÎ² â¥ xâ ch(Î»âÎ²)}) â¤ Ht({y : yt â¥ xâ 2ch(Î»âÎ²)})â¤ Xt([xâ Î»â1/6,â)) (by (3.3.25))â¤ Xt([Ï(Î»),â)) (since xâ Ï(Î») â¥ Î»â1/6 on E1)= Î»â2/3,the last by (3.3.24). Use the above fact that Ht({y : ytâÎ»âÎ² â¥ x â ch(Î»âÎ²)}) â¤ Î»â2/3 onE1 and condition on Ft (recall (3.3.26) with u = 0) to conclude thatP (E1 â© {Î¶tâÎ»âÎ² > Î»âÎ² + Î»â1/2})â¤ E(1(Ht({y : ytâÎ»âÎ² â¥ xâ ch(Î»âÎ²)) â¤ Î»â2/3)P (Zt+Î»â1/2 > 0|Ft))â¤ PÎ»â2/3(XÎ»â1/2(1) > 0)= 1â exp(â2Î»â2/3Î»â1/2)(by (3.3.21))â¤ 2Î»â1/6. (3.3.28)So (3.3.27) and (3.3.28) show thatP (E1 â© {Î¶tâÎ»âÎ² /â [Î»âÎ² , Î»âÎ² + Î»â1/2]}) â¤ 2Î»â1/6 for λ â¥ λ(p, t). (3.3.29)LetM(Ï) = XtâÎ»âÎ² ([xâ ch(Î»âÎ²),â)).If ζ = inf{s â¥ 0 : Xs(1) = 0} is the lifetime of the Feller diffusion Xs(1), then we may apply137(3.3.26) with u = Î»âÎ² to see thatP (Î¶tâÎ»âÎ² â [Î»âÎ², Î»âÎ² + Î»â1/2])= E(EM (ζ â [Î»âÎ² , Î»âÎ² + Î»â1/2]))= E(PM (XÎ»âÎ²+Î»â1/2(1) = 0)â PM (XÎ»âÎ² (1) = 0))= E(exp( â2MÎ»âÎ² + Î»â1/2)â exp(â2MÎ»âÎ²))(by (3.3.21))â¤ E(exp(â2M/(Î»âÎ² + Î»â1/2))2M)[1/Î»âÎ² â 1/(Î»âÎ² + Î»â1/2)]= E(exp(â2M/(Î»âÎ² + Î»â1/2))2M/(Î»âÎ² + Î»â1/2))[Î»â1/2/Î»âÎ² ]â¤ Î»â((1/2)âÎ²) = Î»â1/6+Îµ â¤ Î»âp, (3.3.30)where we have used supxâ¥0 xeâx = eâ1 â¤ 1 in the last line. Combining (3.3.29) and (3.3.30)we arrive atP (E1x,λ) â¤ 3Î»âp for all λ â¥ λ(p, t), x â R.This then implies that for some c3.3.31 = c3.3.31(p, t),P (E1x,λ) â¤ c3.3.31(p, t)Î»âp for all λ â¥ 1, x â R. (3.3.31)Consider next E2 = E2x,λ where for now λ â¥ 1 and of course x â R. Recall thatUs = sup(S(Xs)). On E2, we have Î»â5/6 â¤ Î»âÎµ â¤ δ(t + 1, c) and so by the modulus ofcontinuity (3.2.8),P (E2 â© {Ut+Î»â5/6 â¥ x+ ch(Î»â5/6)}) â¤ P (E2 â© {Ht+Î»â5/6({y : yt â¥ x}) > 0})â¤ PÎ»â1(XÎ»â5/6(1) > 0),where we have used (3.2.13) with δ = 0, and Ht({y : yt â¥ x}) â¤ 1/λ on E in the lastline. Now use (3.3.21) to see that the above equals 1 â exp(â2Î»â1Î»5/6) â¤ 2Î»â1/6, and soconclude thatP (E2 â© {Ut+Î»â5/6 â¥ x+ ch(Î»â5/6)}) â¤ 2Î»â1/6 for all λ â¥ 1, x â R. (3.3.32)138The modulus of continuity also impliesP (E2 â© {Ut+Î»â5/6 â¤ xâ Î»â1/6 â ch(Î»â5/6)})â¤ P (E2 â© {Ht+Î»â5/6({y : yt â¥ xâ Î»â1/6}) = 0})â¤ P (E2 â© {Ht+Î»â5/6({y : yt â¥ Ï(Î»)}) = 0}) (recall xâ Ï(Î») < Î»â1/6 on E2)= PÎ»â2/3(XÎ»â5/6(1) = 0) (by(3.2.13) with δ = 0, and (3.3.24))= exp(â2Î»â2/3Î»â5/6)= exp(â2Î»1/6) â¤ Î»â1/6,the last since λ â¥ 1. The above inequality and (3.3.32) imply thatP (E2 â© {Ut+Î»â5/6 /â (xâ Î»â1/6 â ch(Î»â5/6), x+ ch(Î»â5/6))}) â¤ 3Î»â1/6 âÎ» â¥ 1, x â R.(3.3.33)Differentiate both sides of the scaling relationship in Lemma 3.3.4(a) and so getâ£â£â£ ââxvâ(t, x)â£â£â£ â¤ tâ3/2âGâ²ââ. (3.3.34)If tâ² = t+ Î»â5/6, x1 = xâ Î»â1/6 â ch(Î»â5/6), x2 = x+ ch(Î»â5/6), and λ â¥ λ(p, t), thenP (Utâ² â (x1, x2]) = P (Xtâ²([x2,â)) = 0)â P (Xtâ²([x1,â)) = 0)= eâvâtâ²(x2) â eâvâtâ² (x1) (by (3.3.12))â¤ tâ3/2âGâ²ââ(x2 â x1) (by (3.3.34))= tâ3/2âGâ²ââ(Î»â1/6 + 2ch(Î»â5/6))â¤ 2tâ3/2âGâ²ââÎ»â1/6,where in the last line we used (3.3.25). The above, together with (3.3.33), implies thatP (E2x,λ) â¤ (2tâ3/2âGâ²ââ + 3)Î»â1/6 for all x â R, λ â¥ λ(p, t).This in turn shows that for some c3.3.35(p, t),P (E2x,λ) â¤ c3.3.35(p, t)Î»â1/6 for all x â R, λ â¥ 1. (3.3.35)Combining (3.3.31) and (3.3.35), we derive (3.3.23), as required.An easy consequence of the above is a rate of convergence of vλ to vâ as Î»ââ. Thiswill play an important role in the proof of Theorem 3.1.2 given in the next section.139Proposition 3.3.7. For any 0 < p < 1/6 there is a C3.3.7(p) such thatsupx|vât (x)â vλt (x)| â¤ C3.3.7(p)tâpâ1Î»âp for all λ, t > 0.Proof. Let 0 < p < 1/6. By (3.3.8) and (3.3.12),eâvÎ»t (x) â eâvât (x) = EXδ0 (eâÎ»Xt([x,â))1(Xt([x,â)) > 0))= EXÎ´0(â« â01(0 < Xt([x,â)) â¤ u)eâÎ»uÎ»du)â¤ C3.3.6(p, t)â« â0upeâÎ»uÎ»du (Proposition 3.3.6)= Î(p+ 1)C3.3.6(p, t)Î»âp. (3.3.36)Recalling from (3.3.13) that vât (x) â¤ 2/t, we also haveeâvÎ»t (x) â eâvât (x) â¥ eâvât (x)(vât (x)â vλt (x)) â¥ eâ2/t(vât (x)â vλt (x)). (3.3.37)Combine (3.3.36) and (3.3.37) and set t = 1 to see thatsupx|vâ1 (x)â vλ1 (x)| â¤ e2Î(p+ 1)C3.3.6(p, 1)Î»âp.The required relation is now immediate from the scaling relations (3.3.11) and Lemma 3.3.4(a).Remark. We do not believe p = 1/6 is sharp in any way. Theorem 1.5 of [69] studiessolutions ofâuât=12â2uâx2â u22, u0 = λδ0.In particular this paper shows (via a Feynman-Kac argument) that for some 0 < C(K) â¤C <â,C(K)tâ(1/2)âÎ»0Î»â(2Î»0â1) â¤ uât (x)â uλt (x) â¤ Ctâ(1/2)âÎ»0Î»â(2Î»0â1),where λ0 = λF0 (as in Theorem 3.1.1) and the lower bound is valid for λ â¥ tâ1/2 and|x| â¤ Kât. So naively changing uλ to vλ leads to replacing F = uâ1 with G = vâ1 , and onemight think that (the t dependence is by scaling (3.3.11))vât (x)â vλt (x) â Ctâ2Î»G0 Î»â(2Î»G0 â1) = Ctâ2Î»â1 as Î»ââ, (3.3.38)where â means bounded below and above for perhaps differing positive constants C. This140rate does hold if x = ââ, where (by (3.3.21) and (3.3.12))vât (ââ)â vλt (ââ) =2tâ 2λ2 + Î»tâ¼ 4tâ2Î»â1 as Î»ââ.However the proof in [69] relies on the scaling of uλ, which differs from that of vλ. Moreoverthere is some evidence that the convergence when x â« 0 is slower. In fact a heuristicargument suggests that the correct rate at +â is given by p = G(0) â 1 â (0, 1). The lastupper bound is obvious because G(0) < G(ââ) = 2. For the lower bound on G(0), notethat by (3.3.12) we have G(0) = N0({X1([0,â)) > 0}), so by symmetry,2G(0) â N0({X1([0,â)) > 0} â© {X1((ââ, 0]) > 0}) = N0({X1(1) > 0}) = 2,thus implying that G(0) > 1.3.4 Proof of Theorem 3.1.2We first establish a lower bound on the probability that Lt has positive mass at distancesof orderât away from zero under canonical measure. This follows readily from momentcalculations in [35].Lemma 3.4.1. There is a finite constant C3.4.1 and for all k â¥ 0, positive constantsc3.4.1(k), such that for all t > 0 and k â¥ 0,N0(Lt([kât,â))2) â¤ N0(Lt(R)2) â¤ C3.4.1 t1â2Î»0 , (3.4.1)andN0({Lt([kât,â)) > 0}â£â£Xt > 0) > c3.4.1(k). (3.4.2)Proof. The first claim is immediate from Theorem 3.A(d). The second claim is an easyapplication of the second moment method as we now show. By Theorem 3.A(b) the firstmoment of Lt([kât,â)) isN0(Lt([kât,â))) = CA tâÎ»0 â« 1(âtz â¥ kât)Ï0(z) dm(z) = CA tâÎ»0 â« âkÏ0 dm = c(k) tâÎ»0 ,where c(k) > 0. Thus by the second moment method, we haveN0({Lt(kât,â) > 0}) â¥ N0(Lt(kât,â))2N0(Lt(kât,â)2) â¥(c(k) tâÎ»0)2C3.4.1 t1â2Î»0 =: 2c3.4.1(k) tâ1.141Because Lt = 0 when Xt = 0 and N0({Xt > 0}) = 2/t, this implies thatN0({Lt(kât,â) > 0} â£â£Xt > 0) â¥ c3.4.1(k).We begin the study of Xt near the upper edge of its support. Recall the notation ÏÇ«(t)from (3.1.18) in the Introduction. We first obtain a preliminary upper bound for the massof Xt near ÏÇ«(t).Lemma 3.4.2. Let t, Ç« > 0 and u > 0. ThenN0(â« âÏÇ«(t)âuX(t, x)dx)â¤ eEB0(exp{ââ« t0vÇ«â1s (Bs + u)ds}),where B is a standard Brownian motion under PB0 and vÇ«â1 is as in (3.3.6). If X0 âMF (R) {0}, then EXX0(â«âÏÇ«(t)âuX(t, x)dx)/X0(1) is bounded by the same expression.Proof. As t is fixed we will write Ï Ç« for Ï Ç«(t). We begin by examining N0(â«âÏÇ«âuX(t, x)dx).It is equal toN0(â«1(x+ u > Ï Ç«)X(t, x) dx)= N0(â«1 (Xt([x+ u,â)) < Ç«)X(t, x) dx)â¤ eN0(â«eâÇ«â1Xt([x+u,â))X(t, x) dx).(We note that the above is true when Ï Ç« â R and when Ï Ç« = ââ, in which case Xt(x +u,â) â¤ Ç« for all x â R.) By Theorem 4.1.3 of Dawson and Perkins [15] and translationinvariance, the above is equal toeEB0(exp(ââ« t0uÇ«â1tâs(Ws âWt â u) ds)), (3.4.3)where W is a standard Brownian motion under PB0 and uλs (x) solves (3.3.6) but withuλ0 = Î»1[0,â). Clearly uλt (x) = vλt (âx) (vλt as in (3.3.6)). So if Bs = âWtâs +Wt, a newstandard Brownian motion under PB0 , the above equalseEB0(exp{ââ« t0uÇ«â1tâs(âBtâs â u)ds})= eEB0(exp{ââ« t0vÇ«â1s (Bs + u)ds}). (3.4.4)Consider next PXX0 , for a non-zero initial condition X0. As above, EXX0(â«âÏÇ«+uX(t, x)dx)142equalsEXX0(â«1(Xt([x+ u,â)) < Ç«)X(t, x)dx),which by Theorem 4.1.1 of [15] is bounded byâ«â«1(Xt([x+ u,â)) < Ç«)dNx0dX0(x0) â¤ X0(1)eEB0(exp{ââ« t0vÇ«â1s (Bs + u)ds}).To obtain the left-hand side of the above, we have ignored the contribution to Xt fromparticles unrelated to the individual selected at x by Xt (the quoted theorem in [15] givingthe rigorous justification), and the inequality follows from the bound (3.4.4) and the factthat the above calculation applies, where now W0 = x0, because B remains a Brownianmotion starting at 0.We can now give the proof of Proposition 3.1.5 (restated below for convenience). Thequantity of interest is bounded in terms of the survival probability of an Ornstein-Uhlenbeckprocess Y killed at rate G(Ys), for which we know the lead eigenvalue is â1 by Proposition3.3.5. This leads to the u2 term in upper bound. Proposition 3.3.7 allows us to make theapproximations which lead to the eigenvalue problem.Proposition 3.1.5. There is a non-increasing function, c3.1.5(t), such that for all t, Ç« > 0and u > 0:(a) For any X0 â MF (R), EXX0(â«âÏÇ«(t)âuX(t, x)dx)â¤ c3.1.5(t)X0(1)(u2 â¨ Ç«).(b) N0(â«âÏÇ«(t)âuXt(dx))â¤ c3.1.5(t)(u2 â¨ Ç«).Proof. The results are trivial if u > 1 so we may assume u â¤ 1. Suppose first that1 â¥ u2 â¥ Ç«. By Lemma 3.4.2 it suffices to showEB0(exp{ââ« t0vÇ«â1s (Bs + u)ds})â¤ c3.1.5(t)(u2 â¨ Ç«). (3.4.5)By the scaling relation (3.3.11) the left-hand side of the above equalsEB0(exp(ââ« t01svÇ«â1s1(Bsâs+uâs)ds)). (3.4.6)We define YËs = eâs/2Bes , which defines a stationary Ornstein-Uhlenbeck process on R. Asthis process is reversible with respect to its stationary measure m, Ys = YËâs = es/2Beâsis also a stationary Ornstein-Uhlenbeck process. We denote its expectation by EY . An143exponential time change (s = eâsË) shows that (3.4.6) is equal toEY(exp(ââ« ââ log tvÇ«â1eâsË1(YsË + uesË/2))dsË)(3.4.7)= EYm(exp(ââ« â0vÇ«â1teâsâ²1(Ysâ² + utâ1/2esâ²/2)dsâ²)).The equality follows from changing variables to sâ² = sË + log t and the stationarity of Y .We next truncate the integral and then add and subtract a vâ1 term. This shows that ifp â (0, 1/6), then (3.4.6) is at mostEYm(exp(ââ« log (1/u2)0vÇ«â1teâs1(Ys + utâ1/2es/2))ds)= EYm(exp(â« log(1/u2)0(vâ1 â vÇ«â1teâs1)(Ys + utâ1/2es/2)ds)Ã exp(ââ« log(1/u2)0vâ1(Ys + utâ1/2es/2)ds))â¤ EYm(exp(C3.3.7(p)â« log(1/u2)0(Ç«â1teâs)âpds)Ã exp(ââ« log(1/u2)0vâ1(Ys + utâ1/2es/2)ds)). (3.4.8)The inequality follows by Proposition 3.3.7. Moreover, since u2 â¥ Ç«,â« log(1/u2)0(Ç«â1teâs)âpds = tâp Ç«p[eps/p]log(1/u2)0â¤ tâpp( ǫu2)p â¤ tâpp.This bounds (3.4.6) above byeC3.3.7tâp/pEYm(exp(ââ« log(1/u2)0G(Ys + utâ1/2es/2)ds)), (3.4.9)where C3.3.7 = C3.3.7(p) and we recall G = vâ1 . Define â(s) byâ(s) =â£â£G(Ys)âG(Ys + utâ1/2es/2) â£â£.Gâ² is continuous and has limit 0 at Â±â, thus âGâ²ââ <â. By the Mean Value Theorem,â(s) â¤ âGâ²ââ tâ1/2 ues/2.144Thus (3.4.9) is bounded above byeC3.3.7tâp/pEYm(exp(â« log(1/u2)0â(s) ds)exp(ââ« log(1/u2)0G(Ys) ds))â¤ eC3.3.7tâp/pEYm(exp(âGâ²ââtâ1/2 uâ« log(1/u2)0es/2 ds)exp(ââ« log(1/u2)0G(Ys) ds))â¤ eC3.3.7tâp/p+2âGâ²ââtâ1/2 EYm(exp(ââ« log(1/u2)0G(Ys) ds)). (3.4.10)Let c(t) = eC3.3.7tâp/p+2âGâ²ââtâ1/2 . The remaining term is the probability that an Ornstein-Uhlenbeck process killed at rate G(Ys) survives until time log(1/u2). If ÏG is the lifetimeof this process, we have bounded (3.4.6) byc(t)EYm(exp(ââ« log(1/u2)0G(Ys) ds))= c(t)P Ym(ÏG > log(1/u2))â¤ c(t)CeâÎ»G0 (log(1/u2)) = c3.1.5(t)u2.The inequality follows from (3.2.4) in Theorem 3.2.1(b) and the final equality is by Propo-sition 3.3.5 and setting c3.1.5(t) = Cc(t). This completes the proof when u2 â¥ Ç«. If u2 < Ç«,we have for (b), say,N0(â« âÏÇ«âuXt(dx))â¤ N0(â« âÏÇ«ââÇ«Xt(dx))â¤ c3.1.5(t)Ç«,where the final inequality follows by applying the (uâ²)2 â¥ Ç« case with uâ² = âÇ«. The argu-ment for (a) is the same.We are now ready to give the proof of Theorem 3.1.2. As suggested in the Introduction,the method of proof is to decompose the measure XtâÎ´ into two measures, to the rightand left of Ï Î´(tâ δ). We then show that there is a uniformly positive probability that, themeasure to the right of Ï Î´(tâÎ´) produces positive mass (at time t) in Lt on a set far enoughto the right that the mass from the measure to the left of Ï Î´(tâÎ´) does not interfere with it.Proof of Theorem 3.1.2. First, consider PXX0 . Let (Ft) denote the usual completed right-continuous filtration generated by the associated historical process, H. Fix t > 0. Letδn = 2ân and only consider n so that δn < t/2. We will show that the martingalePXX0(Lt > 0â£â£FtâÎ´n) is bounded below by a positive number a.s. on {Xt > 0}, and so,as it converges to 1{Lt>0} a.s., the latter must be 1 a.s. on {Xt > 0}.145Set Ïn = ÏÎ´n(tâ δn), that is,Ïn = inf{x â R : XtâÎ´n([x,â)) < δn} â¥ ââ.Now invoke the decomposition in (3.2.12) and (3.2.13) with Ï = Ïn and δ = δn. That is, wedefine random measures byXRtâÎ´n(dx) = 1{xâ¥Ïn}XtâÎ´n(dx), XLtâÎ´n(dx) = 1{x<Ïn}XtâÎ´n(dx), (3.4.11)and define measure-valued processes byXËRs (Ï) = XËR,Ïn,Î´ns (Ï) =â«Ï(ytâÎ´n+s)1(ytâÎ´n â¥ Ïn)HtâÎ´n+s(dy),XËLs (Ï) = XËL,Ïn,Î´ns (Ï) =â«Ï(ytâÎ´n+s)1(ytâÎ´n < Ïn)HtâÎ´n+s(dy)XËs(Ï) = XËRs (Ï) + XËLs (Ï) (= XtâÎ´n+s(Ï)).Therefore by (3.2.12) and (3.2.13),Xt = XËÎ´n = XËRÎ´n + XËLÎ´n , where conditional on FtâÎ´n , XËR and XËL are independentsuper-Brownian motions with initial states XRtâÎ´n and XLtâÎ´n , respectively. (3.4.12)Below we will argue conditionally on FtâÎ´n and hence work with this pair of independentsuper-Brownian motions, XËR and XËL, with initial laws XRtâÎ´n and XLtâÎ´n . We apply thecluster decomposition (3.2.6) to each of these super-Brownian motions to concludeXËRÎ´nD=NRâi=1XËR,iÎ´n , XËLÎ´nD=NLâi=1XËL,iÎ´n , (3.4.13)where NR is a Poisson r.v. with rate 2XRtâÎ´n(1)/Î´n, NL is an independent Poisson r.v.with rate 2XLtâÎ´n(1)/Î´n, and, conditional on (NR, NL), the clusters {XËR,iÎ´n: i â¤ NR} are iidwith lawâ«Nx(XÎ´n â ·|Xδn > 0)XRtâÎ´n (dx)/XRtâÎ´n (1) and similarly {XËL,iÎ´n: i â¤ NL} are iidwith lawâ«Nx(XÎ´n â ·|Xδn > 0)XLtâÎ´n(dx)/XLtâÎ´n (1). These last two collections are alsoconditionally independent. (Note also that if XLtâÎ´n = 0, say, then NL = 0 and so there areno clusters to describe.) Let LËRÎ´n , LËLÎ´nand LËÎ´ndenote the boundary local times of XËRÎ´n, XËLÎ´nand XËÎ´n , respectively. By (3.4.12) and applying Theorem 3.2.2 conditionally on FtâÎ´n we146haveLËÎ´n(dx) = 1{XËLÎ´n (x)=0}LËRÎ´n(dx) + 1{XËRÎ´n (x)=0}LËLÎ´n(dx). (3.4.14)We also let LËR,iÎ´n denote the boundary local time of XËR,iÎ´n. If µâ ν denotes product measure,it follows thatPXX0(Lt([Ïn + 3âÎ´n,â)) > 0|FtâÎ´n)= PXX0(LËÎ´n([Ïn + 3âÎ´n,â)) > 0|FtâÎ´n)â¥ PXXRtâÎ´nâ PXXLtâÎ´n(â«1(x â¥ Ïn + 3âÎ´n)1(XËLÎ´n(x) = 0)LËRÎ´n(dx) > 0)â¥ PXXRtâÎ´n(LËRÎ´n([Ïn + 3âÎ´n,â)) > 0)PXXLtâÎ´n (XËLÎ´n([Ïn + 3âÎ´n,â)) = 0), (3.4.15)where in the third line we use (3.4.12) and (3.4.14). Now work on {Ïn > ââ} â FtâÎ´n andconsider the first term in (3.4.15). In this case XRtâÎ´n(1) = δn, and so NR is Poisson withmean 2. Therefore by restricting to {NR = 1} and noting that in this case LËRÎ´n = LËR,1Î´n, wehavePXXRtâÎ´n(LËRÎ´n([Ïn + 3âÎ´n,â)) > 0)â¥ 2eâ2PXXRtâÎ´n(LËRÎ´n([Ïn + 3âÎ´n,â)) > 0|NR = 1)= 2eâ2â« âÏnNx(LÎ´n([Ïn + 3âÎ´n,â)) > 0|Xδn > 0)XRtâÎ´n(dx)/Î´n= 2eâ2â« âÏnN0(LÎ´n([Ïn â x+ 3âÎ´n,â)) > 0|Xδn > 0)XRtâÎ´n (dx)/Î´nâ¥ 2eâ2N0(LÎ´n([3âÎ´n,â)) > 0|Xδn > 0),where the last line again uses XRtâÎ´n(1) = δn on {Ïn > ââ}. Therefore Lemma 3.4.1 and(3.4.15) now imply that on {Ïn > ââ},PXX0(Lt([Ïn + 3âÎ´n,â)) > 0â£â£FtâÎ´n) â¥ 2eâ2c3.4.1(3)PXXLtâÎ´n (XËLÎ´n([Ïn + 3âÎ´n,â)) = 0).(3.4.16)It remains to handle the final probability. We will consider events on which it has auniform lower bound and which will occur infinitely often in n. For K â N, define an eventAK,n â FtâÎ´n byAK,n ={â« â3weâw2/2XtâÎ´n(Ïn â (w â 3)âÎ´n) dw â¤ KâÎ´n}. (3.4.17)(Note that the AK,n depends only on mass to the left of Ïn, and so the measure in the147integral is equal to XLtâÎ´n .) Noting that XtâÎ´n(ââ) = 0, we see that {Ïn = ââ} â AK,n.On AK,n, we have the following lower bound on the probability on the right-hand side of(3.4.16):PXXLtâÎ´n(XËÎ´n([Ïn + 3âÎ´n,â)) = 0) â¥ eâc3.2.3K =: qK . (3.4.18)To prove (3.4.18), first note it is trivial when Ïn = ââ, because in this case XLtâÎ´n = 0. Tosee it when Ïn > ââ we apply Theorem 3.2.3(ii) with R = 3âÎ´n and initial state XLtâÎ´n ,along with translation invariance and the change of variables w = (Ïn +3âÎ´n â x)/âÎ´n, toobtain (on AK,n)PXXLtâÎ´n(XËÎ´n([Ïn + 3âÎ´n,â)) = 0)â¥ exp(â c3.2.3â« Ïnââ(Ïn + 3âÎ´n â x)â2(Ïn + 3âÎ´n â xâÎ´n)3Ã exp(â(Ïn + 3âÎ´n â x)2/2Î´n)XtâÎ´n(x) dx)= exp(âc3.2.31âÎ´nâ« â3w eâw2/2XtâÎ´n(Ïn â (w â 3)âÎ´n) dw)â¥ eâc3.2.3K , (3.4.19)which proves (3.4.18), with the final inequality using the fact that Ï â AK,n.Let ÎK = {AK,n â© {Ïn > ââ} infinitely often in n}. That is,ÎK =ââM=1ânâ¥M(AK,n â© {Ïn > ââ}) . (3.4.20)By (3.4.16) and (3.4.18), for all Ï â ÎK , we havePXX0(Lt([Ïn + 3âÎ´n,â)) > 0â£â£FtâÎ´n) â¥ 2eâ2c3.4.1(3) qK =: pK for infinitely many n.It follows thatlim supnââPXX0(Lt > 0â£â£FtâÎ´n) â¥ lim supnââPXX0(Lt([Ïn + 3âÎ´n,â)) > 0â£â£FtâÎ´n) (3.4.21)â¥ pK a.s. on ÎK .Moreover, PXX0(Lt > 0â£â£FtâÎ´n) is a bounded martingale and converges a.s. to PXX0(Lt >0 | Ftâ). Because sâ Xs is a continuous map, we have Xt = Xtâ and so Xt is measurablewith respect to Ftâ . Moreover, Lt is defined as a measurable functional of Xt (recall148Theorem 3.A(a)). Thus we havePXX0(Lt > 0â£â£FtâÎ´n)â PXX0(Lt > 0 | Ftâ) = 1{Lt>0} a.s. (3.4.22)By (3.4.21), this implies that 1{Lt>0}(Ï) â¥ pK > 0 a.s. on ÎK , and henceLt > 0 almost surely on ÎK . (3.4.23)The final ingredient of the proof is to show that ÎK â {Xt > 0} as K ââ a.s. We proceedby bounding the probability of AcK,n. Using (3.4.17) and Markovâs inequality, we havePXX0(AcK,n) = PXX0(AcK,n â© {Ïn > ââ}) (3.4.24)â¤ 1KâÎ´nEXX0(1{Ïn>ââ}â« â3weâw2/2XtâÎ´n(Ïn â (w â 3)âÎ´n) dw).The first equality follows because AcK,n â {Ïn > ââ}. We proceed by integration by parts.For w > 3, define g(w) byg(w) =1âÎ´nXtâÎ´n([Ïn â (w â 3)âÎ´n, Ïn]) =(â« w3XtâÎ´n(Ïn â (uâ 3)âÎ´n) du).The last expression, which follows from a change of variables, makes it clear thatgâ²(w) = XtâÎ´n(Ïn â (w â 3)âÎ´n).Clearly g(3) = 0. Taking f(w) = weâw2/2 and integrating by parts, on {Ïn > ââ} we haveâ« â3weâw2/2XtâÎ´n(Ïn â (w â 3)âÎ´n) dw=[f(w)g(w)]â3ââ« â3f â²(w)g(w) dw= 0 +1âÎ´nâ« â3(w2 â 1)eâw2/2XtâÎ´n([Ïn â (w â 3)âÎ´n, Ïn]) dw.We substitute this into (3.4.24) and exchange the order of integration (the integrand ispositive) to obtainPXX0(AcK,n) â¤1Î´nKâ« â3(w2 â 1)eâw2/2 PXX0(1{Ïn>ââ}XtâÎ´n([Ïn â (w â 3)âÎ´n, Ïn]))dw.(3.4.25)We now note that the mass term appearing in the integral can be controlled by Proposition1493.1.5. We have for w â¥ 3PXX0(1{Ïn>ââ}XtâÎ´n(Ïn â (w â 3)âÎ´n, Ïn)) â¤ PXX0(â« âÏnââÎ´n(wâ3)XtâÎ´n(x) dx)â¤ c3.1.5(t/2)X0(1)(δn + ((w â 3)âÎ´n)2)= c3.1.5(t/2)X0(1) δn (1 + (w â 3)2).(3.4.26)The second inequality is by Proposition 3.1.5 and our initial assumption that δn < t/2.Using (3.4.26) in (3.4.25), we obtain for n â¥ n0,PXX0(AcK,n) â¤c3.1.5(t/2)X0(1)Kâ« â3(w2 â 1)(1 + (w â 3)2)eâw2/2 dw = c0(t,X0(1))K=:c0K.(3.4.27)This allows us to bound PXX0({Xt > 0} â© ÎcK) as follows:PXX0({Xt > 0} â© ÎcK) = limMââPXX0({Xt > 0} â©[ âân=MAcK,n âª {Ïn = ââ}])â¤ limMââPXX0({Xt > 0} â©[( ânâ¥MAcK,n)â( âân=M{Ïn = ââ})])â¤ limMââPXX0(( ânâ¥MAcK,n)â( âân=M{Ïn = ââ} â© {Xt > 0}))â¤[limMââPXX0(AcK,M)]+ PXX0({Xt(1) > 0} â© {XtâÎ´n(1) â¤ δn i.o.})â¤ c0K. (3.4.28)The second term vanishes because s â Xs(1) is continuous almost surely, and the boundon the first is by (3.4.27). We therefore have thatPXX0({Xt > 0}\u00CEK) â¤ c0K, (3.4.29)and hence for PXX0-almost all Ï â {Xt > 0}, Ï â ÎK for K sufficiently large. Here we alsouse the fact that ÎK is increasing in K. This and (3.4.23) completes the proof that Lt > 0a.s. on {Xt > 0}.The claim that Lt((Ut â δ, Ut)) > 0 almost surely on {Xt > 0} now follows from twoelementary lemmas, the second of which is left as a standard exercise (a variant is knownas Huntâs Lemma).150Lemma 3.4.3. For all δ > 0, almost surely we havelim supnââ1{Lt([Ïn+3âÎ´n,â))>0} â¤ 1{Lt((UtâÎ´,Ut))>0}. (3.4.30)Lemma 3.4.4. Let (Fn)nâN be an arbitrary filtration, Fâ the minimal Ï-algebra containingFn for all n, and let {Yn}nâN be a sequence of random variables such that |Yn| â¤W for alln â N for some integrable W . Thenlim supnE(Yn | Fn) â¤ E(lim supnYn | Fâ).Lemma 3.4.3 is proved at the end of the section. First we see how they completethe proof of Theorem 3.1.2 for PXX0 . Applying (3.4.21) and the Lemmas, with Yn =1{Lt([Ïn+3âÎ´n,â))>0} in Lemma 3.4.4, we have1{Lt((UtâÎ´,Ut))>0} â¥ lim supnââEXX0(1{Lt([Ïn+3âÎ´n,â))>0}|FtâÎ´n)â¥ pK a.s. on ÎK .So by (3.4.29), Lt((Ut â δ, Ut)) > 0 PXX0-a.s. on {Xt > 0}.Proof of Lemma 3.4.3. We may work on {Xt > 0} as both sides are zero if Xt = 0. ByDominated Convergence we havelimnâââ« UtUtâÎ´XtâÎ´n(x)dx =â« UtUtâÎ´X(t, x)dx > 0 on {Xt > 0}.This implies that for n large enough,â« UtUtâÎ´XtâÎ´n(x)dx > 2ân, and soÏn + 3âÎ´n > Ïn > Ut â δ for n large.Therefore for n sufficiently large,Lt([Ïn + 3âÎ´n,â)) â¤ Lt((Ut â Î´,â)) = Lt((Ut â δ, Ut)),and the result follows.This completes the proof of Theorem 3.1.2 under PXX0 . To see that the same holds underN0, we apply the above result with X0 = δ0. We may assume that Xt is defined by theright-hand side of the cluster decomposition (3.2.6). So Xt is a sum of N â¼ Poisson(2/t)independent canonical clusters with law N0(Xt â · |Xt > 0), and N = 1 with probability1512tâ1eâ2/t > 0. In particular we can condition on N = 1, which givesN0(Xt((Ut â δ, Ut)) > 0 for all δ > 0â£â£Xt > 0)= PXδ0(Xt((Ut â δ, Ut)) > 0 for all δ > 0â£â£N = 1) = 1by the result under PXδ0 and the inclusion {N = 1} â {Xt > 0}. Thus the result also holdsunder N0.3.5 LocalizationIn this section we prove Theorem 3.1.3, which states that Lt has positive mass on anyneighborhood of any point in âS(Xt) almost surely. The proof uses both decompositionsfrom Section 3.2.2, Theorem 3.1.2, and the elementary topological fact that if x â âS(Xt),there is a sequence of open âholesâ in the support near x (Lemma 3.5.1 below).Let dH denote the Hausdorff metric on non-empty compact subsets of R. That is,dH(K1,K2) = d0(K1,K2) + d0(K2,K1), where d0(K1,K2) = inf{δ > 0 : K1 â Kδ2} and Kδ2is the set of points which are less than distance δ from K2.Lemma 3.5.1. For x0 â R, x0 â âS(Xt) if and only if there exists two sequences of non-empty open intervals Im and Jm such that dH(Im, {x0}), dH (Jm, {x0}) â 0 as m â â,which satisfy Xt(Â·)|Im = 0 and Xt(Â·)|Jm > 0 for all m.Proof. Let x0 â âS(Xt). A sequence (Jm)âm=1 with the described conditions must existbecause Xt(Â·) is continuous. We know B(x0, 2âm) 6â {Xt > 0} because x0 is not an interiorpoint of {Xt > 0}. So we may choose an open interval Im inside the non-empty open setB(x0, 2âm)â©{Xt > 0}c which is contained in B(x0, 2âm)â©{Xt = 0}, as required. We leavethe converse as an easy exercise.Proof of Theorem 3.1.3. We first work under PXX0 and may assume Xt = Ht({yt â Â·}) whereH is an associated historical process. Let t > 0, q â Q, and δn = 2ân where we may consideronly δn < t. We again use the decomposition (3.2.12), now with Ï = q and δ = δn, that isXËL,q,Î´ns (Ï) =â«Ï(ytâÎ´n+s) 1(ytâÎ´n < q)HtâÎ´n+s(dy)XËR,q,Î´ns (Ï) =â«Ï(ytâÎ´n+s) 1(ytâÎ´n â¥ q)HtâÎ´n+s(dy). (3.5.1)As XËL,q,Î´ns + XËR,q,Î´ns = XtâÎ´n+s, both XËL,q,Î´ns and XËR,q,Î´ns have densities, which we denoteby XËL,q,Î´ns (x) and XËR,q,Î´ns (x). Recall from (3.2.11) that XL,q,Î´ntâÎ´n (A) = XtâÎ´n(A â© (ââ, q))and XR,q,Î´ntâÎ´n (A) = XtâÎ´n(A â© [q,â)) for measurable A â R. Let Ft be the usual right152continuous, completed filtration generated by H. By (3.2.12) and (3.2.13), we haveXt = XËR,q,Î´nÎ´n+ XËL,q,Î´nÎ´n , and conditional on FtâÎ´n , XËL,q,Î´n and XËR,q,Î´n are independentsuper-Brownian motions with initial laws XL,q,Î´ntâÎ´n and XR,q,Î´ntâÎ´n , respectively. (3.5.2)Therefore by Theorem 3.A, for each s > 0, XËR,q,Î´ns and XËL,q,Î´ns each have a boundarylocal time, which we denote by LËR,q,Î´ns and LËL,q,Î´ns , respectively. By (3.5.2) and applyingTheorem 3.2.2 conditionally on FtâÎ´n , we have the following decomposition of Lt:Lt(Ï) =â«Ï(x)1(XËR,q,Î´nÎ´n (x) = 0) dLËL,q,Î´nÎ´n(x) +â«Ï(x)1(XËL,q,Î´nÎ´n (x) = 0) dLËR,q,Î´nÎ´n(x).(3.5.3)Let U q,δn = supS(XËL,q,Î´nÎ´n ). By (3.5.2) and applying Theorem 3.1.2 conditionally on FtâÎ´n ,we see that LËL,q,Î´nÎ´n ((Uq,δn â δ, U q,δn)) > 0 for all δ > 0 almost surely on {XËL,q,Î´nÎ´n > 0}.Taking a union over countable events, this implies that(ân â N,â q â Q, XËL,q,Î´nÎ´n (1) > 0â LËL,q,Î´nÎ´n((U q,δn â δ, U q,δn)) > 0 â δ > 0)PXX0-a.s.(3.5.4)The fact that S(LËR,q,Î´nÎ´n ) â â{x : XËR,q,Î´nÎ´n(x) > 0} implies thatXËR,q,Î´nÎ´n ((ââ, U q,δn)) = 0â LËR,q,Î´nÎ´n((ââ, U q,δn)) = 0 a.s..Therefore by (3.5.3) and (3.5.4),(ân â N,â q â Q,XËL,q,Î´nÎ´n (1) > 0 and XËR,q,Î´nÎ´n((ââ, U q,δn)) = 0 imply (3.5.5)Lt((Uq,δn â δ, U q,δn)) = LËL,q,Î´nÎ´n ((U q,δn â δ, U q,δn)) > 0 âÎ´ > 0)PXX0-a.s.Recall the definitions of h and K(c, δ) from Section 3.2.2 (see (3.2.7)). By the modulus ofcontinuity for S(Ht), (3.2.8), if c > 2, then PXX0-a.a. Ï, there exists δ = δ(c, Ï) > 0 suchthat S(Ht) â K(c, δ). Thus there exists â¦0 â â¦ such that PXX0(â¦c0) = 0 and for all Ï â â¦0,the event in (3.5.5) and S(Ht) â K(3, δ) for some δ(3, Ï) > 0 both hold. Let Ï â â¦0. Leta < b and suppose that (a, b) â© âS(Xt) 6= â . Then there exists x0 â âS(Xt) â© (a, b). ByLemma 3.5.1 there are sequences (Im)âm=1, (Jm)âm=1 of non-empty open intervals convergingto x0 with respect to dH such that Xt(Â·)|Im = 0 and Xt(Â·)|Jm > 0. Suppose Im = (am, bm)and Jm = (dm, em). Since Im, Jm â {x0}, we can consider m large enough so that Im, Jm â(a, b). Without loss of generality we assume that Jm lies to the left of Im, i.e. that em < am.(If Jm lies to the right of Im, a symmetrical argument dealing with the left-hand endpointsLr,δn of the supports of Xr,δn applies).153Let I â²m be the open middle third of Im, i.e., bm â am = lm and I â²m = (am + lm/3, am +2lm/3). Choose q â Qâ© I â²m and n â N large enough so that 3h(δn) < lm/3 and δn < δ(3, Ï).By (3.5.1) and the modulus of continuity,S(XËR,q,Î´nÎ´n ) â (q â 3h(Î´n),â) â (am,â),and henceXËR,q,Î´nÎ´n ((ââ, am]) = 0. (3.5.6)Moreover, because (XËL,q,Î´nÎ´n + XËR,q,Î´nÎ´n)(Â·) = Xt(Â·) > 0 on Jm = (dm, em), and em < am, wehave thatXËL,q,Î´nÎ´n ((dm, am)) > 0. (3.5.7)Furthermore, the modulus of continuity also implies thatS(XËL,q,Î´nÎ´n ) â (ââ, q + 3h(δn)) â (ââ, bm),where the last inclusion holds since 3h(δn) < lm/3 and q â I â²m. The above, together withXt((am, bm)) = 0, implies that S(XËL,q,Î´nÎ´n) â (ââ, am]. This and (3.5.7) imply U q,δn â(dm, am] â (a, b). By (3.5.6) this implies that XËR,q,Î´nÎ´n ((ââ, U q,δn)) = 0 and so by (3.5.7)we may apply (3.5.5) and conclude that Lt((Uq,δn â δ, U q,δn)) > 0 for all δ > 0. SinceU q,δn â (dm, am], choosing δ = (dm â a)/2 > 0 (the last by Jm â (a, b)) gives (U q,δn âÎ´, U q,δn) â (a, b), and hence Lt((a, b)) > 0. This proves the result for PXX0 .To see that the same holds under N0, we proceed as in the proof of Theorem 3.1.2 andcondition that the Poisson number of clusters, N in (3.2.6), is one to getN0((a, b) â© âS(Xt) 6= â â Lt((a, b)) > 0â£â£Xt > 0)= PXδ0((a, b) â© âS(Xt) 6= â â Lt((a, b)) > 0â£â£N = 1) = 1.Thus, under N0 the result holds almost surely on {Xt > 0} for all rational a, b, and henceholds almost surely.154Chapter 4New properties for the density ofthe (α, β)-superprocessSummary. We consider the density Xt(x) of the critical (α, β)-superprocess in Rd with α â(0, 2) and β < αd . A recent result from partial differential equations [7] implies the followingdichotomy: for fixed x â Rd, Xt(x) > 0 a.s. on {Xt 6= 0} if and only if β â¤ Î²â(Î±) := αd+α .We strengthen this by proving that if β < Î²â(Î±) and the density is continuous, which holdsif and only if d = 1 and α > 1 + β, then Xt(x) > 0 for all x â R a.s. on {Xt 6= 0}.In the d-dimensional case when β < αd we then give close to sharp conditions on ameasure µ such that µ(Xt) :=â« Xt(x)µ(dx) > 0 a.s. on {Xt 6= 0}. Our characterizationis based on the size of the support of µ, in the sense of Hausdorff measure and dimension.For s â [0, d], if β â¤ Î²â(Î±, s) = Î±dâs+Î± and supp(µ) has positive xs-Hausdorff measure, thenµ(Xt) > 0 a.s. on {Xt 6= 0}; and when β > Î²â(Î±, s), if µ satisfies a uniform lower densitycondition which implies dim(supp(µ)) < s, then P (µ(Xt) = 0 | Xt 6= 0) > 0.Our methods also give new results for the fractional non-linear equation which is dualto the superprocess, i.e.âtu(t, x) = â(ââ)Î±2 u(t, x) â u(t, x)1+βwith domain (t, x) â (0,â) Ã Rd. The initial trace of a solution ut(x) (see [8]) is a pair(S, ν), where the singular set S is a closed set around which local integrals of ut(x) divergeas t â 0, and ν is a Radon measure which gives the limiting behaviour of ut(x) on Sc ast â 0. For β < αd we characterize the problem of existence of solutions with initial trace(S, 0) in terms of a critical dimension dc = d + α(1 â Î²â1). For S 6= Rd with dim(S) > dc(and in some cases dim(S) = dc) we prove that no such solution exists. When dim(S) < dcand S is the compact support of a measure satisfying a uniform lower density condition, weprove that a solution exists.1554.1 Introduction and statement of resultsIn this chapter we study some path properties of the (α, β, d)-superprocess. For parametersα â (0, 2), β â (0, 1], d â N, the (α, β, d)-superprocess, or simply the (α, β)-superprocesswhen the dimension is fixed, is a strong Markov process taking values inMF (Rd), the spaceof finite measures on Rd equipped with the topology of weak convergence. We will denote itby X = (Xt : t â¥ 0), so that Xt â MF (Rd). The spatial Markov process associated to X is arotationally symmetric α-stable process in Rd and, for β â (0, 1), the branching mechanismis that of a continuous state branching process with (1 + β)-stable branching. When β = 1the superprocess is binary branching and the associated continuous state branching processis Fellerâs branching diffusion.The paths of X live in D([0,â),MF (Rd)), the space of ca`dla`g paths in MF (Rd). TheMarkov transition kernel of X is defined by its Laplace functional, which is characterized viaa dual relationship with a fractional non-linear evolution equation. For a general probabilityspace realizing X with initial value X0 â MF (Rd), we will write (â¦,F , PXX0), and denote theassociated expectation by EXX0 . Let B+b (Rd) = B+b denote the space of bounded, measurablefunctions on Rd. Then for every Ï â B+b ,EXX0(exp(âXt(Ï))) = exp(âX0(uÏt )), (4.1.1)where Xt(Ï) =â«Ï(x)Xt(dx), and uÏt (x) is the unique solution of the evolution equationut(x) = StÏ(x)ââ« t0Stâs(u1+Î²s )(x) ds (4.1.2)for (t, x) â Q := (0,â) Ã Rd. (St)tâ¥0 denotes the transition semigroup of the isotropic(rotationally symmetric) α-stable process in Rd. (See Theorem 4.4.1 of [13] for the existenceand uniqueness of solutions to (4.1.2) and the derivation of (4.1.1).) The generator of theα-stable process is the fractional Laplacian âÎ± = â(ââ)Î±2 . We will use the probabilisticconvention, so that âÎ± corresponds to the α-stable process; the actual âfractionâ associatedto this operator, and the parameter commonly used in the partial differential equation(PDE) literature, is then α2 â (0, 1). In the singular integral formulation, âÎ± is defined asâÎ±f(x) := limÇ«â0+âaÎ±,dâ«Rdf(x)â f(y)|xâ y|d+α ÏÇ«(|xâ y|) dyfor a constant aα,d > 0, whereÏÇ«(r) =ï£±ï£²ï£³1 if r > Ç«,0 if r â [0, Ç«].156The integral equation (4.1.2) corresponds to the fractional PDEâtu = âÎ±uâ u1+β . (4.1.3)When they exist, solutions of (4.1.3) and (4.1.2) generally coincide. In Section 4.2.3 wedefine weak solutions to (4.1.3) (see Definition 4.2.4) and make this correspondence rigorousin certain cases. In addition to studying solutions of (4.1.3) (or (4.1.2)) as a means of provingproperties of X , we also prove novel results concerning the existence and non-existence ofsolutions to (4.1.3) with very singular initial conditions.We are interested in the density of Xt. It is a classical result of Fleischmann [27] that Xtis absolutely continuous if and only if β < αd . This work is concerned only with absolutelycontinuous case, and we restrict to it now.Assumption. For the remainder of this chapter, we assume β < αd .Under this assumption, Xt has a density Xt(x), so that Xt(dx) = Xt(x)dx. A priori, ageneric density is only defined up to Lebesgue-null sets. Much of this work concerns thebehaviour of the density on such sets, so this is not sufficient. In particular, we need theobjectµ(Xt) :=â«RdXt(x)µ(dx) (4.1.4)to be well-defined for µ â MF (Rd). Let B(x, r) denote the closed ball of radius r > 0around x â Rd. We defineX ǫt (x) =Xt(B(x, Ç«))|B(x, Ç«)| , (4.1.5)where |A| denotes the Lebesgue measure of A â Rd. By the Lebesgue DifferentiationTheorem, X ǫt (x) converges as Ç« â 0 for Lebesgue-a.e. x â Rd, and consequentlyXt(x) := lim infÇ«â0X ǫt (x) (4.1.6)is a density for Xt. In fact, more holds.Lemma 4.1.1. For fixed x â Rd, X ǫt (x) â Xt(x) PXX0-a.s as Ç« â 0. Moreover, for µ âMF (Rd), X ǫt (x) â Xt(x) for µ-a.e. x almost surely and µ(X ǫt ) â µ(Xt) in L1(PXX0) asǫ â 0.The above is in fact an abridged version of Lemma 4.2.1, which is proved in Section 4.2.2.Alongside this, as we discuss in Section 4.2.3, the evolution equation has a unique solutionwith initial condition given by a finite measure. Consequently, both sides of (4.1.1) are well-defined when Ï is replaced with a finite measure. The measure case can then be summarized157as follows: (see Lemma 4.2.9 for a precise statement) for µ â MF (Rd), there is a uniquesolution uµt (x) to (4.1.3) on Q such that uµt â µ weakly in the sense of measures as t â 0.The solution uµt satisfiesEXX0(exp(âµ(Xt))) = exp(âX0(uµt )). (4.1.7)In the above, µ(Xt) is defined by (4.1.4) with the density Xt(x) from (4.1.6).Another approach to specifying a canonical version of the density is via a Greenâs func-tion representation for Xt, which is the method used by Fleischmann, Mytnik and Wachtelin [28] and other works. This version of the density is given byXt(x) = X0 â pt(x) +â«(0,t]ÃRdptâs(y â x)M(d(s, y)). (4.1.8)In the above, pt is the transition density of the symmetric α-stable process (see Section 4.2.1)and the measureM is a compensated stable martingale measure associated to X . Althoughwe do not show it here, the density above will agree with the version we use.The canonical measure associated with the (α, β)-superprocess, which we denote N0, is aÏ-finite measure supported on D([0,â),MF (Rd)), the space of ca`dla`g MF (Rd)-valuedpaths. In the construction of superprocesses as scaling limits of discrete spatial branch-ing models, one takes the number of individuals in the population to infinity while theirmasses are simultaneously scaled to 0. The canonical measure N0 is then the âlawâ of thesuperprocess when started with a single (infinitessimal) ancestor at the origin. Likewise, Nxdescribes the superprocess descending from an ancestor located at x â Rd. In Section 4.2.5we describe the relationship between canonical measure and the superprocess with law PXX0 .The following formula will be used quite frequently: for any x â Rd, for t > 0,Nx(Xt 6= 0) = Ut :=(1βt) 1β. (4.1.9)(This is shown, for example, in (5.4.2) of [12].) We remark that Ut is the maximal solutionon (0,â) to the ODE uâ² = âu1+Î². By (4.1.9), when considering Xt under Nx for some fixedt > 0, one is essentially working with a finite measure. This formula is a consequence ofthe fact that the canonical measure also shares a close relationship with the dual evolutionequation. If uÏt (x) is as in (4.1.2) for Ï â B+b , thenNx(1â exp(âXt(Ï))) = uÏt (x). (4.1.10)As with PXX0 , the above relationship can be generalized to include measures when Xt hasa density. If uµt is the unique solution to (4.1.3) with initial condition µ â MF (Rd), (see158Lemma 4.2.9 for details) thenNx(1â exp(âµ(Xt))) = uµt (x). (4.1.11)We now motivate our results with the statement of two theorems. The first is a fun-damental result about superprocesses associated to α-stable spatial motions. The result isoriginally due to Perkins, whose proof of the β = 1 case appeared in [80]. The proof forβ < 1 appears in a more recent work of Li and Zhou [61]. We define supp(µ) to be theclosed support of µ â MF (Rd) and will often refer to it as the support of µ.Theorem 4.A. [Perkins (1990); Li, Zhou (2008)] For X0 â MF (Rd) and t > 0,PXX0(supp(Xt) = Rd or â ) = 1.Similarly, N0(supp(Xt) = Rd | Xt 6= 0) = 1.The superprocesses we study are critical and therefore go extinct almost surely. That is,PXX0(Xt 6= 0 for all t > 0) = 0. (This is analogous to the almost sure extinction of criticalbranching processes, whose spatial analogues have scaling limits given by superprocesses.)Thus, the above statement can be understood to say that, conditioned on non-extinctionat time t > 0, (i.e. conditioned on {Xt 6= 0}) supp(Xt) = Rd a.s.Theorem 4.A is sometimes called instantaneous propagation. This is because, regardlessof the choice of initial measure, Xt has mass âeverywhereâ in Rd. So for X0 = δx, varyingx over Rd has no influence on the support of Xt, which is Rd almost surely on {Xt 6= 0} forany choice of x. The condition supp(Xt) = Rd is equivalent to: Xt(U) > 0 for every openset U â Rd, where Xt(U) = (Xt, 1U ) and 1U is the indicator function of U . So for openU , Xt 6= 0 implies Xt(U) > 0 almost surely. For λ > 0, consider now uλ1U , which is thesolution toâtu = âÎ±uâ u1+β, u0 = λ1U .By (4.1.10),Nx(1â exp(âÎ»Xt(U))) = âuÎ»1Ut (x).Taking Î»ââ with standard limiting arguments, we observe thatlimÎ»ââNx(1â exp(âÎ»Xt(U))) = Nx(Xt(U) > 0).By instantaneous propagation, the right hand side is equal to Nx(Xt 6= 0), which, by trans-159lation invariance, is equal for any choice of x. The implication is thatlimÎ»ââuÎ»1Ut (x) = N0(Xt 6= 0)for all x â Rd. Hence uâ1Ut = limÎ»ââ uλ1Ut is constant in space for each t > 0. In particular,by (4.1.9), for all x â Rd we have uâ1Ut (x) = Ut.Remark 4.1.2. Using the same argument, it follows that N0(Xt(Ï) > 0) = Ut for anymeasurable function Ï â¥ 0 which is positive on a set of positive Lebesgue measure. Inparticular, limÎ»ââ uÎ»Ït = Ut for any such function. These probabilistic results, which webelieve are not widely known in the PDE community, imply the non-existence of positivesolutions to (4.1.3) whose singular sets (see (4.1.24)) have positive Lebesgue measure.The other result motivating ours is a recent result in the PDE literature which, toour knowledge, is not yet widely known in the probability community. It is due to Chen,Ve´ron, and Wang [7]. We first give the result originally stated for PDE, then interpretprobabilistically. Let uλt denote the solution to (4.1.3) with initial data λδ0 and let uât =limÎ»ââ uλt .Let Î²â(Î±) = αd+α . We will generally view d as fixed and therefore omit the dependenceof Î²â(Î±) on d.Theorem 4.B. [Chen, Ve´ron, Wang (2017); Chen, Ve´ron (2019)] Let t > 0 and x â Rd.(a) Let β â¤ Î²â(Î±). Then uât = Ut.(b) Let Î²â(Î±) < β < αd . Then uât satisfiesC1tâ1Î²1 + |tâ 1Î±x|d+Î±â¤ uât (x) â¤ C2tâ1Î² log(e+ |tâ 1αx|)1 + |tâ 1αx|d+α(4.1.12)for constants 0 < C1 < C2.We introduce some terminology: if a solution to (4.1.3), or a limit of solutions is equalto Ut, then we will call the solution (or limit) flat, because these solutions are constant forfixed t > 0. Otherwise it is non-flat. Thus uât is flat when β â¤ Î²â(Î±) and non-flat whenÎ²â(Î±) < β < αd .With the exception of the case that β = Î²â(Î±), this result was proved in [7], while theβ = Î²â(Î±) case was proved by the first two authors of that paper in [8]. Rather surprisingly,depending on the parameters (α, β, d), uât (x) is either flat or has almost the same asymptoticdecay as the heat kernel associated to âÎ±. We interpret the above probabilistically. From160(4.1.11) with µ = λδx, we haveN0(1â exp(âÎ»Xt(x))) = uλt (x),where Xt(x) is the density of Xt at x. Taking Î»ââ yieldsN0(Xt(x) > 0) = uât (x), (4.1.13)from which the following is an immediate consequence of Theorem 4.B.Theorem 4.1.3. Fix t > 0. The following hold under N0 and PXX0 for X0 â MF (Rd).(a) Let β â¤ Î²â(Î±). Then for fixed x â Rd, Xt(x) > 0 almost surely on {Xt 6= 0}. Inparticular, Xt(x) > 0 for Lebesgue-a.e. x â Rd almost surely on {Xt 6= 0}.(b) Let Î²â(Î±) < β < α/d. Then {x : Xt(x) > 0} has finite Lebesgue measure almost surely.Moreover, we haveC1tâ1Î²1 + |tâ 1Î±x|d+Î±â¤ N0(Xt(x) > 0) â¤ C2 tâ 1β log(e+ |tâ 1αx|)1 + |tâ 1αx|d+α(4.1.14)for constants 0 < C1 < C2.Theorem 4.B(a) implies part (a) above as follows: by (4.1.9) and (4.1.13), when β â¤Î²â(Î±) we haveN0(Xt(x) > 0) = N0(Xt 6= 0).Since {Xt(x) > 0} â {Xt 6= 0}, it follows that Nx(Xt(x) = 0 | Xt 6= 0) = 0 and Fubiniâstheorem implies that Xt(x) > 0 for Lebesgue-a.e. x almost surely under N0(Â· | Xt 6= 0). Inpart (b), the fact that {x : Xt(x) > 0} has finite Lebesgue measure under N0 follows fromintegrability of the upper bound in (4.1.14) and Fubiniâs theorem. For PXX0 , the result thenfollows from the cluster representation for the superprocess (see Section 4.2.5).In keeping with the terminology of instantaneous propagation for the behaviour fromTheorem 4.A, we propose to call strong instantaneous propagation the property described forβ â¤ Î²â(Î±). Both results say, in some sense, that Xt has mass âeverywhere.â Instantaneouspropagation describes this on the level of mass on open sets, whereas strong instantaneouspropagation concerns the density at a fixed point.A priori, taken as an immediate consequence of Theorem 4.B, we lack a probabilisticintuition for this result, as Theorem 4.B is proved using analytical methods. In Section 4.3we give a probabilistic proof of part (a). The arguments there are prototypical of otherarguments used later on to prove some of our other results.161Open Problem. Give a probabilistic proof of Theorem 4.1.3(b).We now strengthen one of the conclusions of Theorem 4.1.3 when a continuous versionof the density Xt(Â·) exists. The starting point for this is another dichotomy for the (α, β)-superprocess, which was proved by Fleischmann, Mytnik and Wachtel [28].Theorem 4.C. [Fleischmann, Mytnik, Wachtel (2010)] Fix t > 0 and consider Xt underPXX0 with X0 â MF (Rd).(a) If d = 1 and α > 1 + β, then there is a version of the density Xt(Â·) which is locallyη-Ho¨lder continuous for all η < α1+β â 1.(b) If d > 1, or d = 1 and α â¤ 1 + β, then â1UXt(Â·)ââ = â for all open sets U almostsurely on {Xt 6= 0}.We will refer to the parameter regime d = 1, α > 1 + β as the continuous case. In thiscase it is understood that we will always work with the continuous version of the densityxâ Xt(x), and it is easy to see that this version agrees with the version defined in (4.1.6).The case from Theorem 4.C(b) is then the discontinuous case. For d = 1, in the parameterregime with α > 1 + β and β < Î²â(Î±), the density enjoys both continuity and stronginstantaneous propagation, and we are able to show the following strict positivity result.Theorem 4.1.4. Let d = 1, α > 1 + β and β < Î²â(Î±). Then for t > 0,Xt(x) > 0 for all x â R almost surely on {Xt 6= 0}under both PXX0 and N0.Remark 4.1.5. Strong instantaneous propagation still holds when β = Î²â(Î±), but the proofof the above does not, and this case is left open. Of course, {x : Xt(x) > 0} has finiteLebesgue measure when β > Î²â(Î±) by Theorem 4.1.3(b).This is an interesting property and is perhaps best understood in the context of similarresults for non-negative solutions to stochastic PDE (SPDE). Consider first the stochasticheat equationâtYt(x) = âYt(x) + Yt(x)Î³WË (t, x) (4.1.15)on (0,â) Ã R, where WË (t, x) is space-time white Gaussian noise and γ > 0. It was shownby Mueller in [66] that if γ â¥ 1, then a non-negative solution Yt(x) satisfies Yt(x) > 0 forall (t, x) â (0,â)ÃR. On the other hand, Perkins and Mueller [67] have shown that whenγ < 1, if Y0 has compact support then Yt has compact support for all t > 0 a.s. The case162γ = 12 corresponds to the density of super-Brownian motion, in which case the result wasoriginally due to Iscoe [39].In the SPDE associated to the (α, β)-superprocess with β < 1 in dimension one, the dif-fusion term is replaced with fractional diffusion and the noise is stable rather than Gaussian.In particular, the density Xt(x) of the (α, β)-superprocess solves the SPDEâtXt(x) = âÎ±Xt(x) + Xtâ(x)11+Î² LË(t, x), (4.1.16)where LË(t, x) is a spectrally positive space-time stable noise of index 1 + β. The Greenâsfunction representation (4.1.8) established in [28] can be viewed as a mild form of solu-tion but only applies at fixed times. Alternatively, weak solutions to the SPDE obtainedby replacing âÎ± with â in (4.1.16) were studied in [70]. The key results of that work(Propositions 4.1 and 5.1) could be generalized to construct weak solutions to (4.1.16).In this context, Theorem 4.1.4 is a (fixed time) strict positivity result for a fractionalSPDE with stable noise, and it is the first such result that we are aware of. Furthermore,because {x : Xt(x) > 0} has finite Lebesgue measure when β > Î²â(Î±), it is apparent thatthe interplay of fractional diffusion and stable noise with non-Lipschitz coefficients lead tonon-trivial behaviour which is not seen in Gaussian SPDE like (4.1.15) (c.f. the results from[66] and [67] discussed above).In order to best understand the results that follow, it will be useful to view Theo-rem 4.A and Theorem 4.1.3 from a unified perspective. Let mU denote the Lebesguemeasure restricted to U â Rd. Theorem 4.A states that if U is open (so mU is non-zero),â« Xt(x)mU (dx) > 0 a.s. on {Xt 6= 0}. Thus Xt(Â·) has mass everywhere in Rd at a macro-scopic levelâthe level of open sets. In terms of the PDE (4.1.3), the equivalent statement isthat limÎ»ââ uλmUt (x) is equal to the flat solution Ut. The Lebesgue measure on an open setU is, locally speaking, the most spread out measure on Rd. The least spread out measure,that is the most concentrated measure, is δx for some x â Rd. Theorem 4.1.3 states thatXt(Â·) integrated against the measure δx, i.e. Xt(x), is positive a.s. on {Xt 6= 0} if andonly if β â¤ αd+α , again with an equivalent interpretation that limÎ»ââ uλδx = Ut. In thiscase, Xt(Â·) has mass everywhere at a microscopic levelâat a single point. Taken together,these two results describe the almost sure behaviour of Xt(Â·) on both the most concentratedand least concentrated measures on Rd: δx and mU , respectively. Our results that followinterpolate between these two extremes to describe the almost sure behaviour of µ(Xt) forgeneral measures µ. We are therefore also able to answer some questions about flatnessand non-flatness of solutions to (4.1.3) that are the limits as λ â â of solutions withinitial condition λµ. Indeed, a general principle underlying this work can be summarizedas follows:163µ(Xt) > 0 almost surely on {Xt 6= 0} if and only if limÎ»ââ uλµt = Ut.A condition which quantifies the size of a measure, in the sense of the size of its support,is the following mass distribution property. A measure µ â MF (Rd) satisfies condition (F1)with parameter s â [0, d], or simply (F1)-s, if:(F1)-s For some constant C > 0, for all x â Rd and r > 0,µ(B(x, r)) â¤ Crs.This condition means that µ is spread out in the sense that its support is large, i.e. at leasts-dimensional. In particular, if µ satisfies (F1)-s, then dim(supp(µ)) â¥ s, where dim(A)denotes the Hausdorff dimension of A â Rd. (See Frostmanâs Lemma below.)We view {Xt(x) > 0} as the event that Xt(Â·) charges the measure δx. Our results givea partial answer the question: which measures does Xt(Â·) charge almost surely?Definition 4.1.6. For s â [0, d], let Î²â(Î±, s) = Î±(dâs)+Î± .We remark that Î²â(Î±, 0) = Î²â(Î±), the critical parameter from Theorem 4.1.3. Fur-thermore, Î²â(Î±, d) = 1, which is the critical parameter implicit in Theorem A, since the(α, β)-superprocess has instantaneous propagation for all β â¤ 1. Recall from (4.1.11) thatNx(1â exp(âÎ»µ(Xt))) = uλµt (x).As Î»ââ the left hand side increases to Nx(µ(Xt) > 0) <â.Definition 4.1.7. Let uâµt (x) = limÎ»ââ uλµt (x).We therefore haveNx(µ(Xt) > 0) = uâµt (x). (4.1.17)Since {µ(Xt) > 0} â {Xt 6= 0}, (4.1.9) and (4.1.17) imply thatsupxuâµt (x) â¤ Ut <â. (4.1.18)The results that follow are fixed time results. For Theorems 4.1.8, 4.1.9, 4.1.10(a) and4.1.12, and the discussion of these results, we fix t > 0.Theorem 4.1.8. Suppose that µ â MF (Rd) satisfies (F1)-s for some s â [0, d] and letβ â¤ Î²â(Î±, s).(a) µ(Xt) > 0 a.s. on {Xt 6= 0} under N0 and PXX0 . Equivalently, we have uâµt = Ut.(b) For any ν â MF (Rd) with supp(µ) â supp(ν), ν(Xt) > 0 a.s. on {Xt 6= 0} under N0and PXX0 , and uâÎ½t = Ut.164Thus if µ is spread out in the sense of (F1)-s and β â¤ Î²â(Î±, s), the density chargesµ almost surely when Xt is conditioned on survival. What is more surprising is part (b),which states that it is the closed support of a measure ν rather than its particular proper-ties which ensure that ν(Xt) > 0 almost surely on {Xt 6= 0}. We next show how the aboveleads to a more general result using Frostmanâs Lemma. For S â Rd, let Hs(S) denote thexs-Hausdorff measure of S and recall that dim(S) denotes the Hausdorff dimension of S.Let MF (S) denote the space of finite measures µ with supp(µ) â S.Frostmanâs Lemma. Suppose that S â Rd is Borel. Then Hs(S) > 0 if and only ifâµ â MF (S) satisfying (F1)-s.See Theorem 8.8 in [62] for a proof. The following is an immediate consequence ofTheorem 4.1.8 and Frostmanâs Lemma.Theorem 4.1.9. Let s â [0, d] and suppose β â¤ Î²â(Î±, s). If µ â MF (Rd) satisfiesHs(supp(µ)) > 0, then µ(Xt) > 0 a.s. on {Xt 6= 0} under PXX0 and N0, and uâµt = Ut.Recall that if dim(supp(µ)) = s, then Hsâ²(supp(µ)) = +â for all sâ² < s. In particular,when β < Î²â(Î±, s) the conclusions above hold when dim(supp(µ)) â¥ s. The assertions inTheorems 4.1.8 and 4.1.9 that uâµt = Ut complement Theorem G of [8], which proves asimilar result when s is an integer and the set (in this setting supp(µ)) is a line or hyperplane.Their result is stated in the language of initial traces, which we discuss later. Our criticalparameter Î²â(Î±, s) agrees with the critical parameter of their result.Observe that Î²â(Î±,Î±) = αd . Since Î²â(Î±, s) is increasing in s, it follows that, if β < αd ,then β < Î²â(Î±, s) for all s â¥ α. The first condition is required for the existence of thedensity, and hence if s â¥ α, β < Î²â(Î±, s) holds over the entire parameter set in which weare interested. In other words, whenever the density exists, almost surely it charges anyµ â MF (Rd) such that dim(supp(µ)) â¥ α. (This also complements an observation made in[8].)The results of Theorems 4.1.8 and 4.1.9 are sharp. When β > Î²â(Î±, s), under com-plimentary assumptions on the measure µ, uâµt is non-flat and µ(Xt) = 0 with positiveprobability on {Xt 6= 0}. In order to state this result precisely we introduce the condition(F2). We say that µ â MF (Rd) satisfies property (F2) with parameter s â [0, d], or (F2)-s,if:(F2)-s For some constant C > 0, for all x â supp(µ) and r â (0, 1],µ(B(x, r)) â¥ Crs.165In contrast with (F1), which implies that the mass of µ is spread out in a certain sense,(F2) tells us that the mass of µ is not too spread out in that same sense. In particular,(F2)-s implies that dim(supp(µ)) â¤ s (for example, see Section 8.7 of [32]).Let d(x,S) = infyâS |x â y| denote the distance between x â Rd and a set S â Rd.The restriction s < α in the following is because this is required for (Î²â(Î±, s), αd ) to benon-empty.Theorem 4.1.10. Suppose that µ â MF (Rd) satisfies (F2)-s for some s â [0, α) and hascompact support supp(µ) = S. Let Î²â(Î±, s) < β < αd .(a) For x â Rd and X0 â MF (Rd), Nx(µ(Xt) = 0 | Xt 6= 0) > 0 and PXX0(µ(Xt) = 0 | Xt 6=0) > 0.(b) Nx(µ(Xt) > 0) = uâµt (x) satisfies the following: there are constants C4.1.19 > c4.1.19 >0 such that for all (t, x) â Q,c4.1.19tâ 1β1 + |tâ 1Î±d(x,S)|d+Î±â¤ Nx(µ(Xt) > 0) â¤ C4.1.19[tâ 1Î²â sα â¨ tâ 1β] log(e+ tâ 1αd(x,S))1 + |tâ 1α d(x,S)|d+α.(4.1.19)In particular, Nx(µ(Xt) > 0) vanishes uniformly on {x : d(x,S) â¥ Ï} as t â 0, for all Ï > 0.(c) If, in addition, µ satisfies (F1)-s, then there is a constant c4.1.20 > 0 such that for allx â Rd and t â (0, 1],Nx(µ(Xt) > 0) â¥ c4.1.20[tâ1Î²â sα1 + |tâ 1α [d(x,S) + diam(S)]|d+α]. (4.1.20)(d) For any ν â MF (S), the conclusions of part (a) hold when µ is replaced with ν. Fur-thermore, for constants C4.1.21 > c4.1.21 > 0 we havec4.1.21tâ 1β1 + |tâ 1α d(x, supp(Î½))|d+Î±â¤ Nx(ν(Xt) > 0) â¤ C4.1.21[tâ 1Î²â sα â¨ tâ 1β] log(e+ tâ 1αd(x,S))1 + |tâ 1αd(x,S)|d+α.(4.1.21)Theorems 4.1.9 and 4.1.10 give a sharp picture of the behaviour of Xt(Â·) when integratedagainst an s-dimensional measure. Restrict to the event {Xt 6= 0}. Then we have that whenµ is at least s-dimensional (i.e. Hs(supp(µ)) > 0) and β â¤ Î²â(Î±, s), Xt(Â·) charges µ a.s.,and when µ is at most s-dimensional (by this we mean µ satisfies (F2)-s) and β > Î²â(Î±, s),the probability that Xt(Â·) charges µ has spatial decay similar to an α-stable heat kernel.Theorem 4.A and Theorem 4.1.3 are the special cases of s = d and s = 0, respectively; ourresults cover s â [0, d]. Furthermore, for s â [0, α) there is a non-trivial transition as wevary β over the critical value Î²â(Î±, s). For s â [α, d] the density charges any s-dimensionalset or measure almost surely for all β < αd .166In the flat case, by Theorem 4.1.8(b) and Frostmanâs Lemma we were able to generalizethe results from (F1)-s measures to general measures whose supports have positive Hs-measure (Theorem 4.1.9). In the non-flat case, our most general result is Theorem 4.1.10(d),which holds for all ν whose support is contained in that of an (F2)-s measure and is thereforeat most s-dimensional. It is an attractive open problem to show that this can be extended tomeasures supported on general compact sets of dimension at most s. With the exception ofthe critical case β = Î²â(Î±, s), this would provide a complete characterization of Nx(µ(Xt) >0) for compactly supported µ.Remark 4.1.11. For the special case of β = 1, which requires d = 1 and α â (1, 2) in orderfor the density to exist, we observe that Î²â(Î±, s) < 1 for all s â [0, 1). By Theorem 4.1.10,this implies that for any µ â MF (Rd) with dim(supp(µ)) < 1, the (α, 1)-superprocess in R1fails to charge µ with positive probability on {Xt 6= 0}.To obtain matching upper and lower bounds on Nx(µ(Xt) > 0) as t â 0 in the non-flatcase requires µ to satisfy both (F1)-s and (F2)-s (see Theorem 4.1.10(c)). This is a strongcondition which is sometimes called Ahlfors-David regularity. Still, the properties (F1)-sand (F2)-s are satisfied by many measures which are, in an appropriate sense, uniform oversome s-dimensional set S. First consider the case where s is an integer. If S is a rectifiablecurve in Rd and µ is its length measure, then µ satisfies (F1)-1 and (F2)-1. If S is a surface,or two-dimensional manifold, then its surface measure µ satisfies (F1)-2 and (F2)-2. (Inview of the discussion following Theorem 4.1.8, this implies that for every β < αd , if µ isthe surface or volume measure for a manifold of dimension at least two, then µ(Xt) > 0a.s. when conditioned on survival). For non-integer s, it is known that if S is a self-similarCantor set of dimension s, then its uniform measure µ satisfies (F1)-s and (F2)-s. (Forexample, see the discussion in Section 3 in [63] or Section 1.2 of [10].) Our results can beapplied directly to all of the examples above.Many random sets support measures satisfying (F1) and/or (F2). To illustrate ourdichotomy for the (α, β)-superprocess we consider the range of an independent fractionalBrownian motion with Hurst parameter H â (0, 1) in Rd. We denote this process by(Bt)tâ¥0 and let R(B) = {Bt : t â [0, 1]}. In order to attain both sides of the dichotomy weassume that H > dâ1 â¨ Î±â1, which requires d â¥ 2 and α â (1, 2). (As we discuss below,dim(R(B)) = Hâ1 â§ d. Our assumption guarantees that dim(R(B)) = Hâ1 < α, and wehave Î²â(Î±,Hâ1) < αd .) The paths of Bt are η-Ho¨lder continuous for η â (0,H), (for examplesee Proposition 1.6 of [77]) from which it can be easily shown that the measure µB definedby µB(A) =â« 10 1A(Bt) dt satisfies (F2)-s almost surely (with a random constant C(Ï)) forall s < Hâ1. Hence if β > Î²â(Î±,Hâ1), by Theorem 4.1.10, µB(Xt) = 0 with positiveprobability on {Xt 6= 0}. On the other hand, dim(R(B)) = Hâ1 â§ d. This is a special caseof Theorem 2.1 of [94]. Given our assumption on H, dim(R(B)) = Hâ1 < α. Hence by167Theorem 4.1.9, (and because supp(µB) = R(B)) if β < Î²â(Î±,Hâ1) then µB(Xt) > 0 a.s.on {Xt 6= 0}.Finally, we specialize the flatness and non-flatness results to the continuous case, whichwe recall from Theorem 4.C is when d = 1 and α > 1 + β. It is an elementary consequenceof continuity of Xt(Â·) that µ(Xt) > 0 if and only if Xt(x) > 0 for some x â supp(µ). Theo-rem 4.1.9 and 4.1.10 then imply the following. (Part (b) is mainly included for contrast.)Theorem 4.1.12. Let d = 1, α > 1 + β and s â [0, d].(a) Suppose β â (Î²â(Î±, s), αd ) (which implies s < α) and let S = supp(µ) for a com-pactly supported measure µ â MF (Rd) satisfying (F2)-s. Then Xt(x) = 0 for all x â Swith positive probability on {Xt 6= 0} under PXX0 and N0. Moreover, (4.1.19) holds whenNx(µ(Xt) > 0) is replaced with Nx({Xt(x) > 0 for some x â S}), and if µ satisfies (F1)-s,then so does (4.1.20).(b) Suppose β â¤ Î²â(Î±, s) and let S â Rd be Borel with Hs(S) > 0. Then almost surely on{Xt 6= 0} there is a point x â S such that Xt(x) > 0.Thus far we have generally viewed α as fixed and β as a variable parameter, with thecritical β (i.e. Î²â(Î±, s)) depending on s, the dimension of S or supp(µ). We would also liketo draw attention to another perspective. We define the saturation dimension associatedto the parameters (α, β, d).Definition 4.1.13. For fixed parameters (α, β, d), the saturation dimension dsat isdsat = dsat(α, β, d) = d+ Î±â αβ. (4.1.22)Thus dsat is simply the value of s for which β = Î²â(Î±, s), and therefore it is the criticaldimension of the set or measure as pertains to this problem. It is the maximum dimensionof a set which the density can fail to charge with positive probability: if Hdsat(S) > 0,implying dim(S) â¥ dsat, then the behaviour of Xt is trivial on S the sense that µ(Xt) > 0a.s. on {Xt 6= 0} for every µ â MF (Rd) with supp(µ) = S. The fact that dsat â¤ dcorresponds exactly to instantaneous propagation, i.e. Theorem 4.A. On the other hand,strong instantaneous propagation (i.e. the conclusions of Theorem 4.1.3(a)) corresponds toparameters for which dsat = 0 (where if dsat < 0 we simply define it to be 0). A summaryof some of the results above in these terms is as follows:â¢ If µ â MF (Rd) with Hdsat(supp(µ)) > 0, then Xt(Â·) charges µ almost surely on {Xt 6=0}.â¢ If S = supp(µ) for some µ â MF (Rd) which satisfies (F2)-s with s < dsat (implyingdim(S) < dsat), then for every ν â MF (S), ν(Xt) = 0 with positive probability on{Xt 6= 0}.168We now turn our attention to the PDE (4.1.3) and state some new results concerningthe initial trace theory for positive solutions to this equation. We emphasize that all ofour results apply only when β < αd , which in the PDE literature is sometimes called thesubcritical absorption regime. Following [8], we define the initial trace. For A â Rd, letCc(A) denote the space of continuous, compactly supported functions on A. A positivesolution u to (4.1.3) has initial trace (S, ν), where S â Rd is closed and ν is a Radonmeasure satisfying ν(S) = 0, ifâ¢ For all ξ â Cc(Sc),limtâ0â«Î¾(x)ut(x)dx =â«Î¾dÎ½. (4.1.23)â¢ For every z â S and Ï > 0,limtâ0â«B(z,Ï)ut(x)dx = +â. (4.1.24)The set S is called the singular set of u, whereas Sc is called the regular set. Theorems A andB of [8] give general conditions under which a positive solution to (4.1.3) can be associatedto an initial trace (S, ν). Our results concern the converse problem of determining if thereexists a solution with a given initial trace.The non-fractional analogue of (4.1.3) isâtu = âuâ up. (4.1.25)The initial trace theory for (4.1.25) is well understood. For an analytic approach whichcovers a wide range of parameters p (including p = 1 + β for β â (0, 1]), see Marcus andVe´ron [64]. Because of the dual relationship with super-Brownian motion, the problem isalso amenable to probabilistic analysis. For the case p = 2, Le Gall [55] characterized thepositive solutions to (4.1.25) using the Brownian snake.1The theory for the fractional equation (4.1.3) is more recent and is far from complete.Theorem B hints at this, as it shows that existence of a solution with initial trace ({0}, 0)(sometimes called a very singular solution) depends on the values of the parameters. Aone-point singular set is the smallest non-trivial singular set, and so a consequence of thetheorem is that when β â¤ Î²â(Î±), the singular initial trace problem is trivial in the sense thatif S is non-empty, the solution equals or exceeds Ut (see also Theorem F of [8]). This issuedoes not arise for solutions to (4.1.25) with p â (1, 2] because the associated superprocess1Probabilistic approaches using superprocesses have been very effective in studying the elliptic equationrelated to (4.1.25), âu = up with p â (1, 2]. The p = 2 case has been studied by Le Gall in [54, 56], againusing the Brownian snake. The work of Dynkin and Kuznetsov (e.g. [20, 21]) gives results covering p â (1, 2],which is the entire range for which the superprocess approach applies.169has compact support which is localized near its initial conditions, and hence the event thatthe range of the superprocess does not intersect a given closed set (or that the superprocessdoes not charge a measure supported on that set in an appropriate sense) always has positiveprobability.Our results about flatness and non-flatness of uâµt allow us to advance the theory ofexistence for solutions to (4.1.3) with a given initial trace. At this stage we are largelyconcerned with existence and we do not consider the regularity of solutions. For this reasonwe study weak solutions. A precise definition of a weak solution to (4.1.3) with initial trace(S, ν) is given in Section 4.7. In that section we also restrict our attention to solutionsu(t, x) which are bounded above by Ut, i.e. u(t, x) â¤ Ut. We define U = U(Q) byU = {u : Qâ [0,â) : ut â¤ Ut â t > 0}. (4.1.26)By duality, in particular (4.1.9), (4.1.10) and (4.1.11), U includes all solutions which admit aprobabilistic representation in terms of the (α, β)-superprocess, which includes all solutionsobtained as the limit of solutions with initial data in MF (Rd) or B+b . See also TheoremD of [8], which proves (for classical solutions) that solutions satisfying a mild integrabilityproperty are bounded by Ut and hence belong to U . It remains unresolved if there existpositive solutions to (4.1.3) which do not belong to U ; because they cannot be obtained asthe limits of solutions with truncated function-valued initial conditions, it is unclear howone might construct one.Theorem 4.1.14. (a) (Non-existence) Let S â Rd be closed with Hdsat(S) > 0. If thesingular set of a solution u to (4.1.3) in U contains S, then ut = Ut. In particular, ifS 6= Rd there is no solution to (4.1.3) in U with initial trace (S, ν) for any Radon measureν.(b) (Existence) Suppose that S â Rd is compact and there exists µ â MF (Rd) satisfying(F2)-s with s < dsat such that S = supp(µ). Then there exists a weak solution to (4.1.3) inU with initial trace (S, 0). The solution is given by limÎ»ââ uλµt (x) = Nx(µ(Xt) > 0) andsatisfies (4.1.19).The problem of existence for solutions with initial trace (S, 0) is therefore characterizedby the saturation dimension. This generalizes Theorem B(a) (and Theorem F of [8]), whichcan be viewed as the special case of the above when dsat = 0. When dsat = 0 there is notransition: all non-empty singular sets fall into the non-existence regime.Organization of the chapter. The rest of the chapter is organized as follows. Sec-tion 4.2 provides background information and preliminary results; the key subsections areSection 4.2.2, which covers the density of Xt and µ(Xt), and Section 4.2.3, which discusses170solutions to (4.1.3) with measure-valued initial data and extends the dual relationship toinclude finite measures. In Section 4.3 we give a probabilistic proof of Theorem 4.1.3(a), andin Section 4.4 we prove Theorem 4.1.4. Sections 4.5 and 4.6 cover, respectively, flatness andnon-flatness of uâµt for general measures µ, in particular Theorems 4.1.8 and 4.1.10. Finally,we discuss solutions to (4.1.3) with non-empty singular sets and prove Theorem 4.1.14 inSection 4.7.4.2 Preliminaries4.2.1 Transition densitiesWe denote by pt(x) the fundamental solution (or heat kernel) to the fractional heat equationon Rd. That is, pt(x) is the solution toâtu = âÎ±u, u0 = δ0on Q = (0,â) Ã Rd. The semigroup (St)tâ¥0 which we have already introduced is a convo-lution semigroup with kernel pt, i.e.StÏ(x) = Ï â pt(x) =â«Ï(y)pt(xâ y)dy.We note that for µ â MF (Rd), Stµ(x) can be defined in the same way with no difficulty.The kernel pt is radial and radially decreasing. In a slight abuse of notation, for Ï > 0 wewill sometimes write pt(Ï) to mean pt(x), where |x| = Ï.(St)tâ¥0 is the transition semigroup of the (rotationally) symmetric α-stable process, sowe have the following: if W is a symmetric α-stable process with law and expectation PWxand EWx , respectively, when started at x, then for an appropriate class of functions (e.g.bounded and measurable),EWx (Ï(Wt)) = StÏ(x).In particular, pt is also the transition density of W , that isPWx (Wt â dy) = pt(y â x)dy.The symmetric α-stable process is self-similar, which is reflected by the scaling property forpt:pt(x) = tâ dα p1(tâ 1αx).Finally we recall the asymptotic decay of the transition density. There are universal con-171stants 0 < c4.2.1 < C4.2.1 <â such thatc4.2.1tâdÎ±1 + |tâ1/Î±x|d+Î± â¤ pt(x) â¤ C4.2.1tâdÎ±1 + |tâ1/Î±x|d+Î± . (4.2.1)It is occasionally useful to write the above asc4.2.1(tâdÎ± â§ t|x|d+Î±)â¤ pt(x) â¤ C4.2.1(tâdÎ± â§ t|x|d+α), (4.2.2)where in order to do so one may have to adjust the constants. Without loss of generalitywe will fix c4.2.1 and C4.2.1 so that both bounds hold.4.2.2 The density of the (α, β)-superprocessIn this section we give an overview of the density of Xt. The main purpose is to show thatwe can take a version of the density which is defined almost surely at any fixed point x â Rdand that with this version the quantity µ(Xt) from (4.1.4) is well-defined almost surely. Wealso discuss the regularity properties of the density in the continuous regime.Recall from (4.1.5) and (4.1.6) that for x â Rd and Ç« > 0,X ǫt (x) =Xt(B(x, Ç«))|B(x, Ç«)|andXt(x) := lim infÇ«â0X ǫt (x).Since Xt is absolutely continuous, x â Xt(x) is a density for it by the Lebesgue differ-entiation theorem (for example see Theorem 3.21 of [30]). Because we are interested inthe behaviour of the density at fixed points and more generally on Lebesgue null sets, werequire more than the standard a.e.-convergence of X ǫt (Â·) to Xt(Â·). The desired conditionsare shown to hold in the next lemma.For Ï : Rd â R and Ç« > 0, let ÏÇ«(x) := Ç«âdÏ(Ç«â1x).Lemma 4.2.1. Let X0 âMF (Rd) and t > 0 and consider Xt under PXX0 .(a) For every x â Rd, X ǫt (x)â Xt(x) a.s. and in L1 as Ç« â 0.(b) For every µ â MF (Rd), X ǫt (x)â Xt(x) for µ-a.e. x almost surely and µ(X ǫt )â µ(Xt)in L1, where µ(Xt) =â« Xt(x)µ(dx). Moreover, we haveEXX0(µ(Xt)) = µ(StX0). (4.2.3)(c) If Ï : Rd â [0,â) satisfies â« Ï = 1 and supx |Ï(x)|(1 + |x|)d+δ < â for some δ > 0,172then the conclusions of part (a) and (b) hold when X ǫt is replaced with Xt â ÏÇ«.We prove this lemma at the end of the section. Part (c) is included because it is some-times useful to approximate Xt(x) using convolutions with general kernels. A particularlyuseful example is when Ï = p1, in which case ÏÇ«1Î±= pǫ.We now turn our attention to the Ho¨lder regularity of the density in the continuousregime, i.e. d = 1 and α > 1 + β (c.f. Theorem 4.C). The following was proved in [28].Theorem 4.D. [Fleischmann, Mytnik and Wachtel (2010)] Let d = 1 and α > 1 + β. LetX0 â MF (Rd) and t > 0. Under PXX0 , there is a continuous version Xt(Â·) of the densitysuch that for every η < ηc =α1+β â 1, Xt(Â·) is locally Ho¨lder of index η, i.e.supx1,x2âK,x1 6=x2|Xt(x1)â Xt(x2)||x1 â x2|η <â for all compact K â R.Furthermore, the value ηc is optimal in that, for any η > ηc, with probability one, for anyopen U â R,supx1,x2âU,x1 6=x2|Xt(x1)â Xt(x2)||x1 â x2|η =â whenever Xt 6= 0.As we have noted in the introduction, when d = 1 and α > 1 + β, the density we definein (4.1.6) is the same as the continuous version. The statement in the above implies thefollowing. Let η < α1+β â 1 and X0 â MF (R). ThenPXX0-a.s., for all compact K â R, there exists C(K, η, Ï) > 0 such that for all x, y â K,|Xt(x)â Xt(y)| â¤ C(K, η, Ï)|x â y|η. (4.2.4)We conclude the section with the proof of Lemma 4.2.1. The proof uses a classical resultconcerning absolute continuity of the laws of superprocesses. As with Theorem 4.A, theresult originates in [24], where it was proved for binary branching superprocesses. The prooffor (1 + β)-stable branching superprocesses appears in [61]. We do not state the result infull generality, but refer the reader to Theorems 1.1 and 2.2, respectively, of [24] and [61].For any X0, XË0 â MF (Rd) and t > 0, we haveThe laws of Xt under PXX0 and PXXË0 are mutually absolutely continuous. (4.2.5)We use the above when XË0 is a translation of X0. In particular, for x â Rd suppose thatXË0 = X0 + x, where for a measure µ we define µ + x by (µ + x)(A) =â«1A(y â x)µ(dy).Applying (4.2.5), the laws of Xt under PXX0 and PXX0+x are mutually absolutely continuous.173On the other hand, by translation invariance of the superprocess, the law of Xt under PXX0+xis equal to the law of Xt â x under PXX0 . Consequently, we have that for x â Rd and t > 0,The laws of Xt and Xt + x are mutually absolutely continuous under PXX0 . (4.2.6)We will also use the following moment bound.Lemma 4.2.2. Let 0 < θ < β and Ï â¥ 0. ThenEXX0(Xt(Ï)1+Î¸) â¤ 1 + C[â« t0X0(Stâs((SsÏ)1+Î²))ds + X0((StÏ)1+Î²)]for a constant C = C(α, d, θ) > 0.The α = 2 case of this result appears as Lemma 2.1 of [71]. The same argument used in[71] also works in the α â (0, 2) when one replaces the heat semigroup with the fractionalheat semigroup, so we omit the proof and continue with the proof of Lemma 4.2.1. Thearguments used to bound moments below are also borrowed from [71].Proof of Lemma 4.2.1. Fix t > 0 and X0 â MF (Rd). Let Ï : Rd â [0,â) satisfyâ«Ï = 1and supx |Ï(x)|(1 + |x|)d+δ <â for some δ > 0 and recall that ÏÇ«(x) = Ç«âdÏ(Ç«â1x). Theseconditions include the case where Ï is the normalized indicator function of the unit ball.A priori this is the case we are interested in, but we carry out the analysis for the generalcase as this is required for part (c).Let X Ç«,Ït (x) = Xt â ÏÇ«(x). Since Xt is absolutely continuous almost surely, we havePXX0 -almost surely, X Ç«,Ït (x) converges for Lebesgue-a.e. x â Rd as Ç« â 0. (4.2.7)If Ï is the normalized indicator function of the unit ball, the above is simply Lebesgueâsdifferentation theorem. Otherwise, the result follows from Theorem 8.15 of [30].We now show that the convergence holds pointwise almost surely. Suppose there existsx0 â Rd such that PXX0({XÇ«,Ït (x0) does not converge as Ç« â 0}) > 0. By (4.2.6) this impliesthat PXX0({XÇ«,Ït (x) does not converge as Ç« â 0}) > 0 for all x â Rd. However, by Fubiniâstheorem this implies that the set of points for which X Ç«,Ït (x) does not converge has positiveLebesgue measure a.s., which contradicts (4.2.7). Hence we must have thatX Ç«,Ït (x) converges PXX0 -a.s. as Ç« â 0 for all x â Rd. (4.2.8)Thus we have shown the almost sure convergence stated in part (a). Before proving the L1convergence and part (b), we show how absolute continuity also implies the a.s. convergencein part (c). When Ï is the normalized indicator function of a unit ball, we denote X Ç«,Ït (x)174by X ǫt (x). Let Ï be any other function satisfying the conditions as above. If there existsx0 â Rd such that PXX0(lim infÇ«â0X ǫt (x0) 6= lim infÇ«â0 XÇ«,Ït (x0)) > 0, then (4.2.6) implies thatthe probability is positive when we replace x0 with x for all x â Rd. By Fubiniâs theorem,lim infÇ«â0X ǫt (x) 6= lim infÇ«â0X Ç«,Ït (x) on a set of positive Lebesgue measure almost surely.Since lim infÇ«â0 X ǫt (Â·) and lim infÇ«â0X Ç«,Ït (Â·) are both densities for Xt, this is a contradiction.Hence lim infÇ«â0X ǫt (x) = lim infÇ«â0 X Ç«,Ït (x) a.s. for every x â Rd. The left hand side issimply equal to Xt(x), and so for all Ï satisfying the stated conditions, (4.2.8) can beimproved toX Ç«,Ït (x)â Xt(x) PXX0-a.s. as Ç« â 0 for all x â Rd,which proves that the almost sure convergence in (a) holds for all Ï from (c). Furthermore,the above implies that the limit of X Ç«,Ït (x) does not depend on the choice of Ï. Given this,we can work with general Ï and prove the remaining claims of (a) and (b) simultaneouslywith (c).In order to establish that X Ç«,Ït (x) â Xt(x) in L1, we will show Lp-boundedness ofthe quantity X Ç«,Ït (x) for p = 1 + θ with θ < β. To do so we apply Lemma 4.2.2. LetÏxÇ« (Â·) = ÏÇ«(Â· â x) and note that X Ç«,Ït (x) = Xt(ÏxÇ« ). For θ < β, we haveEXX0((X Ç«,Ït (x))1+θ) â¤ 1 + C[â« t0X0(Stâs((SsÏxÇ« )1+β))ds + X0((StÏxÇ« )1+β)]. (4.2.9)Sinceâ«ÏxÇ« = 1, we can view ÏxÇ« as the density of a random variable, which we will denoteY . Thus for s > 0,(SsÏxÇ« )(z)1+β =[â«ps(y â z)ÏxÇ« (y)dy]1+β= EY (ps(Y â z))1+Î²â¤ EY (ps(Y â z)1+Î²)=â«ps(y â z)1+Î²ÏxÇ« (y)dy, (4.2.10)We also haveps(y â z)1+β = ps(y â z)ps(y â z)β â¤ Cps(y â z)sâdÎ²Î±by (4.2.2). Hence by (4.2.10) and the above,X0(Stâs((SsÏxÇ« )1+β)) â¤ CsâdÎ²Î±â«â«â«ptâs(z âw)ps(y â z)ÏxÇ« (y) dy dz X0(dw)= CsâdÎ²Î±â«â«pt(w â y)ÏxÇ« (y) dy X0(dw)â¤ CX0(1)sâdÎ²Î± tâdÎ± . (4.2.11)175Since β < αd , the power of s is integrable over s â [0, t], which implies that the first termin the square brackets in (4.2.9) is uniformly bounded in Ç« â (0, 1]. The second termcan be handled the same way and is in fact easier to control. Thus EXX0((XÇ«,Ït )1+θ) isuniformly bounded for Ç« â (0, 1], which implies that the family {X Ç«,Ït (x) : Ç« â (0, 1]} isuniformly integrable. Since X Ç«,Ït (x) converges almost surely to Xt(x), this implies that theconvergence is also in L1(PXX0).We now fix µ â MF (Rd) and show L1 convergence of µ(X Ç«,Ït ) to µ(Xt). Note that thebound in (4.2.11) does not depend on x. Thus, rather than consider L1+θ-boundednessof Xt(ÏxÇ« ) for fixed x with respect to PXX0 , we can fix µ â MF (Rd) and consider L1+θ-boundedness of (Ï, x) â Xt(ÏxÇ« )(Ï) with respect to PXX0 â µ. Because the bounds areuniform in x and µ is finite, the argument requires no modification. Since Xt(ÏxÇ« )â Xt(x)a.s. for every x, it follows that Xt(ÏxÇ« ) â Xt(x) in L1(PXX0 â µ) (as a function of (Ï, x)).This implies that µ(X Ç«,Ït )â µ(Xt) in L1.It remains to show the moment formula (4.2.3). First, let Ï : Rd â [0,â) be boundedand measurable. The mean measure formula for superprocesses givesEXX0(Xt(Ï)) = X0(StÏ). (4.2.12)Now fix µ â MF (Rd). We haveµ(X Ç«,Ït ) =â«â«ÏÇ«(xâ y)Xt(dy)µ(dx) =â«â«µÇ«(x)Xt(dx) = Xt(µǫ),where µǫ = µ â ÏËÇ« with ÏËÇ«(Â·) = ÏÇ«(â·). Since µǫ is a bounded and measurable function, by(4.2.12) the above impliesEXX0(µ(X Ç«,Ït )) = X0(Stµǫ).Because µ(X Ç«,Ït ) â µ(Xt) in L1(PXX0), the left hand side converges to EXX0(µ(Xt)) as Ç« â 0.We have Stµǫ â¤ µ(1)tâ dα for all Ç« > 0, so the right hand side converges to X0(Stµ) byDominated Convergence. Since X0(Stµ) = µ(StX0), this proves (4.2.3) and the proof iscomplete.4.2.3 The fractional PDE and µ(Xt)In this section we extend the duality between Xt and solutions to the evolution equa-tion (4.1.2). Recall from (4.1.1) that in its basic form, the dual relationship states thatEXX0(exp(âXt(Ï)) = exp(âX0(uÏt )), where for bounded and measurable Ï â¥ 0, uÏt is theunique solution to (4.1.2) with initial data Ï. The purpose of this section is to extend thisrelationship to allow Ï to be replaced with a finite measure when β < αd . We also introduceweak solutions to (4.1.3) and some of their properties.176The integral equation (4.1.2) is a mild form of the PDE (4.1.3). We will work withweak solutions. Recall that Q = (0,â) Ã Rd. Let C1,2c (Q) denote the space of compactlysupported functions on Q which are once and twice continuously differentiable in time andspace, respectively. For T > 0, let QT = (0, T ]ÃRd and QÂ¯T = [0, T ]ÃRd, and let C1,2b (QÂ¯T )denote the space of bounded functions on QÂ¯T with bounded, continuous derivatives up toorder one in time and order two in space. For p â¥ 1, we let Lploc(Q) denote the space offunctions Ï such thatâ«K |Ï|p <â for every compact K â Q.Definition 4.2.3. A function u : Qâ R is a weak solution to (4.1.3) if (t, x) â u(t, x) iscontinuous, u â L1+βloc(Q), andâ«Q(u(t, x)[ââtÎ¾(t, x)ââÎ±Î¾(t, x)]) + u(t, x)1+βξ(t, x) dxdt = 0 (4.2.13)for all ξ â C1,2c (Q).For measure-valued initial data, the PDE problem of interest isï£±ï£²ï£³ âtu = âÎ±uâ u1+β for (t, x) â Q,u0 = µ(4.2.14)for µ â MF (Rd). In the following, recall that by convergence in MF (Rd) we mean weakconvergence of measures.Definition 4.2.4. For µ â MF (Rd), we say that u : Qâ R+ is a weak solution to (4.2.14)if it is a weak solution to (4.1.3), u â L1(QT ) â© L1+β(QT ) for all T > 0, and ut â µ inMF (Rd) as t â 0.Proposition 4.2.5. For µ â MF (Rd), there exists a unique weak solution u : Qâ [0,â)to (4.2.14). Moreover, for T > 0 we haveâ«QT(u(t, x)[ââtÎ¾(t, x)ââÎ±Î¾(t, x)]) + u(t, x)1+βξ(t, x) dxdt=â«RdÎ¾(0, x)µ(dx) ââ«Rdu(T, x)ξ(T, x)dx (4.2.15)for all ξ â C1,2b (QÂ¯T ).We will denote the unique solution to (4.2.14) by uµt (x) or uµ(t, x). The above isproved in Theorem 1.1 of [7]. The authors of that work use a slightly different definitionwhich incorporates (4.2.15). However, a short argument which we omit shows that given asolution in the sense of Definition 4.2.4, (4.2.15) holds for all ξ â C1,2b (QÂ¯T ). The definition177of a solution used in [7] also does not require continuity, but it can be verified that uµt (x)is jointly continuous, for example using its correspondence with the solution to the integralequation given in Lemma 4.2.7(b) below. (The proof of Lemma 4.2.7(b) requires only asmall modification if one does not assume a priori that uµt (x) is continuous.) The followingstability result holds as a consequence of Theorem 1.1 of [7].Lemma 4.2.6. (a) If µ â MF (Rd) and (µn)nâ¥1 is a sequence of measures such that µn â µin MF (Rd), then uµnt (x)â uµt (x) locally uniformly in Q.(b) The map µâ uµt is increasing.Solutions of (4.2.14) are bounded above by solutions of the homogeneous fractional heatequation with the same initial data. Again by Theorem 1.1 of [7], we haveuµt (x) â¤ Stµ(x) for (t, x) â Q, (4.2.16)where (t, x)â Stµ(x) is the unique solution to âtv = âÎ±v on Q with v0 = µ (see Theorems3.1 and 5.1 of [3]).Lemma 4.2.7. Let µ â MF (Rd). (a) The integral equationut(x) = Stµ(x)ââ« t0Stâs(u1+Î²s )(x) ds, (t, x) â Q (4.2.17)has a unique, non-negative, jointly continuous solution in L1(QT )â©L1+Î²(QT ) for all T > 0.(b) The weak solution to (4.2.14) with initial data µ and the solution to (4.2.17) are equal.We will therefore use the notation uµt (x) and uµ(t, x) to refer to the unique solution to(4.2.14) and (4.2.17). With the exception of continuity, part (a) of the above is proved inLemma A.2 of [27], and continuity can be shown by a direct calculation which we omit.Proof of Lemma 4.2.7(b). Let µ â MF (Rd) and let u(t, x) = uµ(t, x), the unique weaksolution to (4.2.14). For T > 0, x0 â Rd and n â N, we take ξ(t, x) = pT+nâ1ât(x â x0) âC1,2b (Q) in (4.2.15). Since pt solves âtpt(x) = âÎ±pt(x), we haveâ«QTu(t, x)1+Î²pT+nâ1ât(xâ x0) dxdt=â«RdpT+nâ1(xâ x0)µ(dx)ââ«Rdu(T, x)pnâ1(xâ x0)dx. (4.2.18)For all n â¥ 1, âpT+nâ1ââ â¤ CTâdÎ± by (4.2.2), and so by bounded convergence the firstterm in the second line converges to STµ(x0). The second term converges to u(T, x0) bycontinuity of u. Now consider the first line. Let Ç« > 0. For n â¥ 1 and t â¤ T â Ç«,178âpT+nâ1âtââ â¤ CÇ«âdÎ± . Since u â L1+β(QT ), we can apply Dominated Convergence toobtain thatlimnâââ« TâÇ«0â«Rdu(t, x)1+Î²pT+nâ1ât(xâ x0) dxdt =â« TâÇ«0â«Rdu(t, x)1+Î²pTât(xâ x0) dxdt.On the other hand, by (4.2.16) there is a constant K > 0 such that u(t, x)1+β â¤ K for allx â Rd and t â [T â Ç«, T ]. Consequently, we havelimÇ«â0supnâ¥1â£â£â£â£â« TTâÇ«â«Rdu(t, x)1+Î²pT+nâ1ât(xâ x0) dxdtâ£â£â£â£ = 0.Combining everything above, we take nââ in (4.2.18) to conclude that for all (T, x0) â Q,u(T, x0) = STµ(x0)ââ«QTu(t, x)1+Î²pTât(xâ x0) dxdt,and hence u is equal to the solution of (4.2.17).Remark 4.2.8. Along the same lines as the above, it can be shown that if u(t, x) is abounded weak solution to (4.1.3) and u(t, x) â Ï(x) a.e. as t â 0 for Ï â B+b , thenu(t, x) = uÏ(t, x), the solution to (4.1.2) with u0 = Ï. This also implies that such a solutionis unique and has the probabilistic representation u(t, x) = Nx(1âexp(âXt(Ï))) by (4.1.10).We now extend the dual relationship with the (α, β)-superprocess to measures.Lemma 4.2.9. Let µ â MF (Rd). Then for x â Rd and t > 0,Nx(1â exp(âµ(Xt))) = uµt (x), (4.2.19)and for X0 â MF (Rd),EXX0(exp(âµ(Xt))) = exp(âX0(uµt )). (4.2.20)Proof. We give the proof under PXX0 and note that it follows by essentially the same ar-gument for the canonical measure. (One can restrict to the event {Xt 6= 0} because1 â exp(âµ(Xt)) = 0 on {Xt = 0}, which allows us to treat Nx as a finite measure.)Fix µ â MF (Rd) and define µn = Snâ1µ. Then µn is smooth, bounded and positive, so by(4.1.1) we haveEXX0(exp(âXt(µn))) = exp(âX0(uµnt )). (4.2.21)Since µn â µ in MF (Rd) as n â â, by Lemma 4.2.6(a) it follows that uµn â uµ locally179uniformly. In particular, uµnt â uµt pointwise. By (4.2.16) and (4.2.2), we have|uµnt (x)| â¤ Cµ(1)tâdÎ±for all n â¥ 1 and x â Rd. Hence X0(uµnt ) â X0(uµt ) by Dominated Convergence, andconsequentlylimnââ exp(âX0(uµnt )) = exp(âX0(uµt )). (4.2.22)Consider now the left hand side of (4.2.21). Expanding (Xt, µn), we haveXt(µn) = Xt(Snâ1µ) =â«Snâ1µ(x)Xt(dx) =â«Snâ1Xt(x)µ(dx).We have used the symmetry of Snâ1 . Note that Snâ1Xt(x) = Xtâpnâ1(x) is an approximationof the density which satisfies the conditions of Lemma 4.2.1(c), and hence Xt(µn)â µ(Xt)in L1(PXX0). In particular, this implies the convergence of the left hand side of (4.2.21) toEXX0(exp(âµ(Xt)). Combined with (4.2.22), this implies (4.2.20) and completes the proof.Remark 4.2.10. Note that (4.2.19) immediately implies that µâ uµt is increasing, whichwe have already stated as Lemma 4.2.6(b). We will refer to this monotonicity in initialconditions of solutions to (4.1.3) as the comparison principle. Given Remark 4.2.8, thecomparison principle also holds for bounded weak solutions with initial data in B+b .Solutions to (4.1.3) satisfy a useful scaling property. For λ > 0 and µ â MF (Rd),one can verify directly using elementary methods and uniqueness of solutions to problem(4.2.14) the following formula:uλµ(t, x) = λαγuµ(·/λβγ)(λαβγt, λβγx), (4.2.23)where γ = 1Î±âÎ² , and the measure µ(·/λβγ) is the dilation of µ defined byµ(·/λβγ)(A) = µ(A/λβγ) =â«1(Î»Î²Î³x â A)µ(dx)for measurable A â Rd. This leads to a very useful expression when we scale out the timevariable t to obtain an expression involving a solution at time 1; in particular, we haveuµt (x) = tâ 1β utÎ±âÎ²Î±Î² µ(Â·/tâ 1α )(1, tâ1Î±x). (4.2.24)180It follows that uâµt (x) = limÎ»ââ uλµt (x) satisfiesuâµt (x) = tâ 1β uâµ(Â·/tâ 1α )(1, tâ1Î±x). (4.2.25)4.2.4 A Feynman-Kac formulaWe now state a Feynman-Kac formula for some functions related to solutions of (4.1.2).First, for Ï,Ï â B+b , we formally definezÏ,Ï(t, x) =ââÇ«uÏ+Ç«Ï(t, x)â£â£â£Ç«=0= limÇ«â0uÏ+Ç«Ï(t, x)â uÏ(t, x)Ç«, (4.2.26)where uÏ(t, x) is taken to be the solution to (4.1.2) with u0 = Ï. By (4.1.10) and theDominated Convergence Theorem, it follows that the derivative with respect to Ç« exists andzÏ,Ï(t, x) = Nx(Xt(Ï) exp(âXt(Ï))).Recall that EWx denotes the expectation associated to an α-stable process Wt with W0 = x.Lemma 4.2.11. (a) For Ï,Ï â B+b ,zÏ,Ï(t, x) = EWx(Ï(Wt) exp(â(1 + Î²)â« t0uÏs (Ws)βds)). (4.2.27)(b) For Ï â B+b and λ > 0, ââÎ»uÎ»Ït (x) exists for all (t, x) â Q andââÎ»uÎ»Ït (x) = EWx(Ï(Wt) exp(â(1 + Î²)â« t0uÎ»Ït (Ws)βds))(4.2.28)In particular, for (t, x) â Q and Î > 0 we haveuÎÏt (x) =â« Î0ââÎ»uÎ»Ït (x) dλ. (4.2.29)A proof of part (a) is implicit in the proof of Theorem 6.3.1 in [12], where the repre-sentation (4.2.27) is established by showing that zÏ,Ï(t, x) solves a certain linear evolutionequation. To see that (b) follows from (a), it suffices to consider zÎ»Ï,Ï(t, x) for Ï â B+b andλ > 0.4.2.5 Cluster decompositionsThe connection between the (α, β)-superprocess and its canonical measure is via a clusterrepresentation in which the superprocess is given by a Poisson superposition of clusters181whose intensity is canonical measure. To make this precise, for X0 â MF (Rd) we defineNX0(Â·) =â«Nx(Â·)X0(dx).Let Î(Â·) be a Poisson point process on D([0,â),MF (Rd)) with intensity NX0 . Then theprocessXt(Â·) =ï£±ï£²ï£³â«Î½t(Â·)Î(dÎ½) if t > 0,X0(Â·) if t = 0is an (α, β)-superproces with law PXX0 . This is a consequence of Theorem 4.2.1 of [18]. Forfixed t > 0 the above implies thatXt D=Nâi=1X it , (4.2.30)whereD= indicates equality of distribution. In the above, N is a Poisson random variablewith mean X0(1)(Î²t)â1Î² and the X it are iid random measures with distribution NX0(Xt â· |Xt 6= 0). This representation gives us a convenient way to compare path propertiesunder PXX0 and Nx. In particular, suppose we realize Xt under PXδx via (4.2.30). Since theprobability that N = 1 is positive, we can condition on this event, and it follows thatNx(Xt â · |Xt 6= 0) = PXδx (Xt â · |N = 1). (4.2.31)Consequently, for measurable A âMF (Rd) we haveIf PXδx (Xt â A | Xt 6= 0) = 1, then Nx(Xt â A | Xt 6= 0) = 1. (4.2.32)4.3 The density at a fixed pointIn the introduction, we noted that Theorem 4.1.3 is equivalent to Theorem 4.B, which isproved analytically. In this section we give a probabilistic proof of Theorem 4.1.3(a), andtherefore of Theorem 4.B(a). We define uλt (x) as the solution to (4.2.14) with initial measureλδ0. Then by translation invariance of the equation (4.1.3) and (4.2.20), for x â Rd we haveEXX0(exp(âÎ»Xt(x))) = exp(ââ«uÎ»t (y â x)X0(dy))(4.3.1)We define uât = limÎ»ââ uλt and observe that, by taking Î»ââ in (4.3.1),PXX0(Xt(x) = 0) = exp(ââ«uât (y â x)X0(dy)). (4.3.2)182The main purpose of this section is to show the following.Proposition 4.3.1. If β â¤ Î²â(Î±) = αd+α , then for fixed x â Rd, Xt(x) > 0 a.s. on {Xt 6= 0}under PXX0 and N0. Hence uât = Ut.In fact, this result is a consequence of the more general Theorem 4.1.8, which we provein Section 4.5. However, we state and prove it separately for a few reasons. First, whileTheorem 4.1.8 (and several other results) concern the behaviour of µ(Xt) for certain familiesof measures, the measure δx is of particular interest because it corresponds to the densityat a fixed point. The other reason is that, while the method used to prove Proposition 4.3.1and Theorem 4.1.8 is largely the same, the proof of the latter involves technicalities thatdo not arise in the former. We therefore opt to include the simpler proof in the case whichis particularly interesting.Recall that we will sometimes write uλ(t, x) to denote uλt (x). We note the particularform that the scaling relationship (4.2.24) takes for the family uλ(t, x). Since δ0(·/r) = δ0,it follows thatuλ(t, x) = tâ1Î² utÎ±âÎ²Î±Î² λ(1, tâ1Î±x) (4.3.3)Consequently, uâ(t, x) satisfiesuâ(t, x) = tâ1Î² uâ(1, tâ1Î±x). (4.3.4)The following lemma gives a lower bound for uλ1 (for λ â¥ 1) in terms of the heat kernelp1 of the symmetric α-stable process and holds for all 0 < β <αd . The statement of thisresult for uâ1 appeared in [7], where it was called Lemma 5.3. Here we give a probabilisticproof.Lemma 4.3.2. There is a constant c4.3.2 = c4.3.2(α, β, d) > 0 such that for all λ â¥ 1,uλ1(x) â¥ c4.3.2p1(x).In particular, uâ1 (x) = N0(X1(x) > 0) â¥ c4.3.2p1(x).Proof. For λ > 0 and Ç« > 0, consider uÎ»pÇ«(t, x), the unique solution to (4.1.2) with u0 = Î»pÇ«.By Lemma 4.2.11(b), we haveââÎ»uÎ»pÇ«(1, x) = EWx(pǫ(W1) exp(â(1 + Î²)â« 10uÎ»pÇ«(1â s,Ws)β ds)).183By (4.2.16), uÎ»pÇ«s â¤ Î»SspÇ« = λps+ǫ, which implies thatââÎ»uÎ»pÇ«(1, x) â¥ EWx(pǫ(W1) exp(â(1 + Î²)â« 10Î»Î²p1âs+Ç«(x+Ws)Î² ds))â¥ EWx (pǫ(W1)) exp(âÎ»Î²(1 + Î²)â« 10p1âs+Ç«(0)Î² ds).In the second line we have used the fact that p1âs+Ç« is radially decreasing, hence p1âs+Ç«(Ws) â¥p1âs+Ç«(0), and removed the exponential from the expectation because it no longer dependson W . We note that EWx (pǫ(W1)) = S1pǫ(x) = p1+Ç«(x) from the semigroup property. Using(4.2.2) and changing variables in the integral, we then haveââÎ»uÎ»pÇ«(1, x) â¥ p1+Ç«(x) exp(âÎ»Î²(1 + Î²)â« 10Cβ4.2.1(s+ Ç«)â dβα ds)Because β < αd , the integral remains bounded as Ç« â 0. It follows that for a constant C > 0,for all Ç« â (0, 1],ââÎ»uÎ»pÇ«(1, x) â¥ p1+Ç«(x) exp(âCÎ»Î²).In particular, (4.2.29) and the above imply thatupǫ(1, x) â¥ c0p1+Ç«(x)for a constant c0 > 0. By Lemma 4.2.6(a), the left hand side converges to u1(1, x) asǫ â 0, and the right hand side converges to p1(x). Thus we have u1(1, x) â¥ c0p1(x). SinceÎ»â uλ(1, x) is increasing, this implies the result.By (4.3.4) and Lemma 4.3.2, one obtains thatuât (x) â¥ c4.3.2tâ1Î² p1(tâ 1αx). (4.3.5)For fixed x 6= 0, by (4.2.2) we then havelim inftâ0uât (x)tâ 1β+ d+αα> 0.It is therefore immediate that limtâ0 uât (x) = â for all x â Rd when β < Î²â(Î±). It doesnot give the same conclusion when β = Î²â(Î±), and in neither case is it immediate that uâtis flat.Let (Ft)tâ¥0 denote the standard right-continuous filtration associated to X = (Xt : t â¥0). For a PXX0-integrable function f(X ), to denote its conditional expectation we will omitthe sub- and superscripts and simply write E(f(X )|F). The (one-dimensional) Markov184property for X is then expressed asE(f(Xt+s) | Fs)(Ï) = EXXs(Ï)(f(Xt)).Because X is ca`dla`g, PXX0(Xt = Xt+ ât > 0) = 1. The following lemma gives almostsurely left continuity at a fixed time.Lemma 4.3.3. Fix t > 0 and X0 â MF (Rd). Almost surely under PXX0 there is no discon-tinuity of s â Xs at time t, and hence Xs â Xt in MF (Rd) as s â t. Moreover, for anyopen or closed ball B, limsât Xs(B) = Xt(B) almost surely.Proof. Fix t > 0. The claim is a consequence of Lemma 1.6 of [28]. Part (a) of that lemmastates that the discontinuities of t â Xt are described by a jump measure N(d(s, x, r)),and the form of the compensator of N given in part (b) implies that a.s. there is no jumpat time t. Hence Xs â Xt weakly as s â t. For an open or closed ball B, the fact thatXs(B) â Xt(B) as s â t follows from weak convergence and the fact that Xt(âB) = 0 (byabsolute continuity).Proof of Proposition 4.3.1. Let X0 â MF (Rd), x â Rd and t > 0. Our method is to showthat PXX0(Xt(x) = 0,Xt 6= 0) = 0. This implies that PXX0(Xt(x) > 0,Xt 6= 0) = PXX0(Xt 6= 0).At the end of the proof we discuss the case for canonical measure and show that uât = Ut.Recall that B(x, r) = {y â Rd : |yâx| â¤ r}. From Theorem 4.A, we have supp(Xt) = Rda.s. on {Xt 6= 0}. In particular, PXX0 -a.s. on {Xt 6= 0} we have Xt(B(x, 1)) > 0. Thus itsuffices to show thatPXX0({Xt(x) = 0} â©AÎº) = 0 (4.3.6)for every κ > 0, where we define the eventAκ = {Xt(B(x, 1)) â¥ κ}.Let κ > 0 and consider the event A2κ. By Lemma 4.3.3, limsâtXs(B(x, 1)) = Xt(B(x, 1)) â¥2Îº. We let δn = 2ân and note that the previous statement implies that XtâÎ´n(B(x, 1)) â¥ κfor n sufficiently large (depending on Ï). That is, if we define Eκn byEκn = {XtâÎ´n(B(x, 1)) â¥ κ},thenfor a.e. Ï â A2κ, âN = N(Ï) such that Ï â â©ân=N(Ï)EÎºn . (4.3.7)185In particular, this implies thatlimnââPXX0(A2Îº â© (Eκn)c) = 0. (4.3.8)Consider the conditional probability P (Xt(x) = 0 | FtâÎ´n ). Applying the Markov property,we obtainP (Xt(x) = 0 | FtâÎ´n) = PXXtâÎ´n (Xδn(x) = 0)= exp(ââ«uâÎ´n(y â x)XtâÎ´n(dy)),where the second equality uses (4.3.2). We now bound above by ignoring all the mass ofXtâÎ´n outside of B(x, 1). This givesP (Xt(x) = 0 | FtâÎ´n ) â¤ exp(ââ«B(x,1)uâÎ´n(y â x)X1âÎ´n(dy))â¤ exp(âc4.3.2â«B(x,1)Î´â 1βn p1(Î´â 1αn (y â x))X1âÎ´n(dy)),where the second inequality uses (4.3.5). Since p1 is radially decreasing, the minimum valueit can attain in the integral above is p1(2Î´â 1αn ), and p1(2Î´â 1αn ) â¥ c1δd+ααn for some c1 > 0 by(4.2.2). We thus obtain that for c2 = c1 · c4.3.2 > 0,P (Xt(x) = 0 | FtâÎ´n ) â¤ exp(âc2Î´â 1β+ d+ααn X1âÎ´n(B(x, 1))).Now suppose that Ï â Eκn. Then X1âÎ´n(B(x, 1)) â¥ κ, and henceFor PXX0 -a.e. Ï â Eκn, we havePXX0(Xt(x) = 0 | FtâÎ´n )(Ï) â¤ exp(âc2Îº Î´â 1β+ d+ααn). (4.3.9)First suppose that β < Î²â(Î±) = αd+α . In this case, the exponent of δn in (4.3.9) is negative186and so the right hand side of (4.3.9) converges to 0 as nââ. We thus obtain thatPXX0({Xt(x) = 0} â©A2Îº)â¤ EXX0(P ({Xt(x) = 0} â©EÎºn | FtâÎ´n))+ PXX0(A2κ â© (EÎºn)c)â¤ PXX0(Eκn) exp(âc2Îº Î´â 1β+ d+ααn)+ PXX0(A2κ â© (EÎºn)c)â 0 as nââ,where we have used (4.3.8) and (4.3.9). We have therefore shown thatPXX0({Xt(x) = 0} â©A2Îº) = 0.This suffices to prove the result, so the proof is complete for β < αd+α .Now suppose that β = αd+α . Here we use a martingale argument. The process P (Xt(x) >0 | FtâÎ´n ) is a bounded martingale with respect to the increasing sequence of Ï-algebras{FtâÎ´n}ân=1. By the martingale convergence theorem it follows thatlimnââP (Xt(x) > 0 | FtâÎ´n )(Ï) = P (Xt(x) > 0 | Ftâ)(Ï) (4.3.10)for PXX0 -a.e. Ï, where FtâÎ´n â Ftâ := Ï(Xs : 0 â¤ s < t). We now show that the right handside of the above is equal to 1(Xt(x) > 0)(Ï) almost surely. By (4.1.6) we haveXt(x) = lim infÇ«â0Xt(B(x, Ç«))|B(x, Ç«)| = lim infÇ«â0Xtâ(B(x, Ç«))|B(x, Ç«)| a.s.,where Xtâ(B(x, Ç«)) = limsât Xs(B(x, Ç«)) exists and equals Xt(B(x, Ç«)) a.s. for all Ç« > 0 byLemma 4.3.3. In a slight abuse of notation, let us denote by Xtâ(x) the quantity on the righthand side of the above. Then Xtâ(x) = Xt(x) almost surely and Xtâ(x) is Ftâ-measurable.We therefore have, for PXX0 -a.e. Ï,P (Xt(x) > 0 | Ftâ)(Ï) = P (Xtâ(x) > 0 | Ftâ)(Ï)= 1(Xtâ(x) > 0)(Ï)= 1(Xt(x) > 0)(Ï).Hence (4.3.10) implies thatlimnââP (Xt(x) > 0 | FtâÎ´n )(Ï) = 1(Xt(x) > 0)(Ï) PXX0 -a.s. (4.3.11)187Since β = αd+α , the exponent of δn in (4.3.9) is zero, and hence the right hand side of theinequality in (4.3.9) is bounded above by exp(âc2Îº) =: c(κ) < 1. From (4.3.7), (4.3.9) and(4.3.11), we can fix Ï outside a null set such that (i) Eκn occurs for sufficiently large n onA2κ, (ii) P (Xt(x) > 0 | FtâÎ´n )(Ï) â 1(Xt(x) > 0)(Ï) as n â â, and (iii) for all n â¥ 1, ifÏ â Eκn we haveP (Xt(x) = 0 | FtâÎ´n )(Ï) â¤ c(κ) < 1.This third condition is equivalent toP (Xt(x) > 0 | FtâÎ´n )(Ï) â¥ 1â c(κ) > 0.From (i) and (iii), P (Xt(x) > 0 | FtâÎ´n)(Ï) â¥ 1 â c(κ) for n sufficiently large. From (ii) weconclude that 1(Xt(x) > 0)(Ï) â¥ 1 â c(κ) > 0, and hence Xt(x) > 0. We have thereforeshown that PXX0({Xt(x) = 0} â© A2κ) = 0, that is (4.3.6) holds, for all κ > 0, which provesthe result. Hence the proof is complete for the case β = αd+α , and we are done.Having shown that PXX0(Xt(x) > 0 | Xt 6= 0) = 1, the result under N0 then followsby (4.2.32). In particular, we obtain that N0(Xt(x) > 0) = N0(Xt 6= 0), and hence thatuât (x) = Ut.4.4 Strict positivity of the densityIn this section we prove Theorem 4.1.4, which states that the density is strictly positiveunder certain conditions in the continuous case. In particular, in dimension one (d = 1)with α > 1 + β (continuity) and β < Î²â(Î±) (strong instantaneous propagation), we haveXt(x) > 0 for all x â R almost surely on {Xt 6= 0}at a fixed time t > 0. The proof of the result hinges in part on the following result, whichgives an exponential rate of decay for the left tail of the density conditional on non-negligiblenearby mass. Its proof shares many ideas with the proof of Proposition 4.3.1.The following holds for general dimensions d â N. We denote BR = {x â Rd : |x| â¤ R}.Lemma 4.4.1. Let β < αd+α . Let R â¥ 1 and t â (0, 1]. There is a constant c4.4.1 > 0which depends only on (α, β, d) such that for any X0 â MF (Rd),PXX0(Xt(x) â¤ tÎ±âÎ²Î±Î²)â¤ e · exp(âc4.4.1X0(BR)Rd+Î±tâq)for all x â BR, where q = 1β â d+αα > 0.Proof. Fix t â (0, 1], R â¥ 1 and X0 â MF (Rd). Let λ = tâÎ±âÎ²Î±Î² . One can verify directly188that1(Xt(x) â¤ Î»â1) â¤ exp (1â λXt(x)) .Using the above and applying (4.3.1), we havePXX0(Xt(x) â¤ Î»â1) â¤ e Â·EXX0 (exp (âÎ»Xt(x)))= e · exp(ââ«uÎ»(t, y â x)X0(dy))â¤ e · exp(ââ«BRuÎ»(t, y â x)X0(dy)). (4.4.1)In the last line we simply disregard the mass of µ outside of BR. Next we obtain a lowerbound on the integrand in the above. From (4.3.3), we haveuλ(t, y â x) = tâ 1β uÎ»Â·tÎ±âÎ²Î±Î²(1, tâ1Î± (y â x))= tâ1Î² u1(1, tâ1Î± (y â x))â¥ c4.3.2tâ1Î² p1(tâ 1α (y â x)).The second line uses the fact that Î»tÎ±âÎ²Î±Î² = 1 and the third follows from Lemma 4.3.2.Finally, for all x, y â BR, we use (4.2.2) and the above to obtain thatuλ(t, y â x) â¥ c tâ 1β+ d+Î±Î±Rd+Î±âx, y â BRfor a constant c > 0 depending only on (α, β, d). Using the above in (4.4.1), we obtain thatPXX0(Xt(x) â¤ Î»â1) â¤ e · exp(âcX0(BR) tâ 1β+ d+ααRd+α).Since λ = tâÎ±âÎ²Î±Î² , this completes the proof.Besides the above, the other main ingredient in the proof of Theorem 4.1.4 is Ho¨ldercontinuity of Xt(x), which we discussed in Section 4.2.2. In particular we will use (4.2.4).As can be seen from the proof, the actual index of Ho¨lder continuity is irrelevant. Anypositive index works.Proof of Theorem 4.1.4. Let d = 1 and α > 1 + β. Let X0 â MF (Rd). By scaling it issufficient to consider the time t = 1. We will show that X1(x) > 0 for all x â [âR,R]PXX0-a.s. for every R â¥ 1, and hence that X1(x) > 0 for all x â R.Fix R â N. As in the proof of Proposition 4.3.1, we will use instantaneous propagation.189In particular, by Theorem 4.A we havePXX0(X1([âR,R]) > 0 | X1 6= 0) = 1.Let ER,κ = {X1([âR,R]) â¥ κ}. By the above, it suffices to show thatX1(x) > 0 for all x â [âR,R] a.s. on ER,κ (4.4.2)for all κ > 0. We fix κ > 0 and consider the event ER,2κ. For a sequence {Î´n}nâN ={2âÎ³n}nâN, with γ > 0 to be specified later, we define events BR,κn byBR,κn = {X1âÎ´n([âR,R]) â¥ κ}.By Lemma 4.3.3, limsâ1 Xs([âR,R]) = X1([âR,R]) â¥ 2κ a.s. on ER,2κ, and hence for a.e.Ï â ER,2κ there is s0(Ï) < 1 such that Xs([âR,R]) â¥ κ for all s â [s0(Ï), 1]. Hence BR,κnoccurs for sufficiently large n, that is,ER,2κ â {BR,κn eventually}, (4.4.3)where{BR,κn eventually} =ââN=1âân=NBR,Îºn .For n â N, let În denote the set of dyadic lattice points at scale 2ân, i.e. În = 2ânZ.Recall that R â N. We then let ÎRn = [âR,R] â© În, that isÎn = {âR+ k2ân : k = 0, 1, . . . , 2R2n}.Next, we defineFn = Fn(R) = {X1(x) > Î´Î±âÎ²Î±Î²n âx â ÎRn }. (4.4.4)The first step of our proof is to show thatPXX0(ER,2κ â© {F cn i.o.}) = 0, (4.4.5)where i.o. is short for infinitely often, meaning{F cn i.o.} =ââN=1âân=NF cn.190We now show that (4.4.5) holds. By (4.4.3), we havePXX0(ER,2κ â© {F cn i.o.}) â¤ PXX0({BR,κn eventually} â© {F cn i.o.})Suppose that Ï is in the event on the right hand side of the above; then (i) there is N(Ï)such that Ï â BR,κn for all n â¥ N(Ï), and (ii) for any m â N, there is n > m so that Ï â F cn.Together, (i) and (ii) imply that for any m â¥ N(Ï), there is n > m such that Ï â BR,κn â©F cn.That is, the above event is a sub-event of {BR,κn â© F cn i.o.}. HencePXX0(ER,2κ â© {F cn i.o.}) â¤ PXX0({BR,κn â© F cn i.o.})= PXX0( ââN=1âân=NBR,Îºn â© F cn)= limNââPXX0( âân=NBR,Îºn â© F cn)â¤ limNâââân=NPXX0(BR,Îºn â© F cn). (4.4.6)We bound the probabilities arising in the final term using Lemma 4.4.1. We condition onF1âÎ´n and note that BR,κn â F1âÎ´n . By the Markov property, we haveP (BR,κn â© F cn | F1âÎ´n) = 1(BR,κn )PXX1âÎ´n (Fcn).â¤âxâÎRn1(BR,Îºn )PXX1âÎ´n(X (δn, x) â¤ Î´Î±âÎ²Î±Î²n).By the definition of BR,κn , in the above we need only compute the probability for such X1âÎ´nas satisfy X1âÎ´n([âR,R]) â¥ κ, in which case we can apply Lemma 4.4.1 for each x â ÎRn .From this we obtain (recall that |ÎRn | = 2R2n)P (BR,κn â© F cn | F1âÎ´n) â¤ C1(R) 2n exp(âc4.4.1ÎºR1+Î±Î´âqn),where C1(R) = 2Re. Recall that we have chosen δn = 2âÎ³n, and so, substituting the aboveinto (4.4.6), we obtainPXX0(ER,2κ â© {F cn i.o.}) â¤ limNâââân=NC1(R) 2n exp(âc4.4.1ÎºR1+Î±2Î³qn)= 0.Thus we have shown that (4.4.5) holds, implying that {Fn eventually} occurs a.s. on ER,2κ.191Recalling the definition of Fn from (4.4.4) and that δn = 2âÎ³n, it therefore holds thatFor a.e. Ï â ER,2κ, there is N(Ï) â N such that for all n â¥ N(Ï),X1(x) > 2âÎ³Î±âÎ²Î±Î²n for each x â ÎRn . (4.4.7)Next we use the Ho¨lder continuity of Xt(Â·). Let 0 < η < α1+β â 1. By (4.2.4) withK = [âR,R], for a random constant C2(R) = C2(R, η, Ï) > 0,|X1(x1)â X1(x2)| â¤ C2(R)|x1 â x2|η for all x1, x2 â [âR,R]. (4.4.8)Having chosen η, we can now choose a corresponding value of γ. Let γ = Î±Î²Î±âÎ²Î·2 . Then by(4.4.7), for n sufficiently large,X1(x) > 2âÎ·2n for each x â ÎRn . (4.4.9)Let y â [âR,R]. We define [y]n = min{x â ÎRn : x â¥ y}. That is, if y 6â ÎRn , then [y]n thepoint in ÎRn nearest to y on the right; if y â ÎRn , then [y]n = y. Note that |y â [y]n| < 2ânfor all y â [âR,R] by the definition of ÎRn . Hence by (4.4.8),supyâ[âR,R]|X1(y)â X1([y]n)| â¤ C2(R) 2âÎ·n. (4.4.10)By the triangle inequality, for y â [âR,R],X1(y) â¥ X1([y]n)â |X1(y)â X1([y]n)|.Note that {[y]n : y â [âR,R]} = ÎRn . Taking the infimum of the above over [âR,R] andapplying (4.4.10), we obtain thatinfyâ[âR,R]X1(y) â¥(infyâ[âR,R]X1([y]n))â supyâ[âR,R]|X1(y)â X1([y]n)|â¥ infxâÎRnX1(x)â C2(R) 2âÎ·n.By (4.4.9), for all sufficiently large n we therefore haveinfyâ[âR,R]X1(y) â¥ 2âÎ·2n â C2(R) 2âÎ·n= 2âÎ·2n(1â C2(R)2âÎ·2n).By taking n to be large enough in comparison to C2(R), the right hand side is positive.192This proves that the density is strictly positive on [âR,R] a.s. on ER,2κ. Hence (4.4.2)holds and the proof is complete.4.5 Almost sure charging of (F1)-s measures whenβ â¤ Î²â(Î±, s)In this section we prove that, under some conditions on α and β, µ(Xt) > 0 almost surelyon {Xt 6= 0} for certain measures µ, which is equivalent to Nx(µ(Xt) > 0) = uâµt = Ut.More precisely, this section contains the proof of Theorem 4.1.8(a). We recall the Frostmancondition (F1) for a measure: for s â [0, d], µ â MF (Rd) satisfies (F1)-s if(F1)-s For some constant C, for all x â Rd and r > 0,µ(B(x, r)) â¤ Crs.Theorem 4.1.8(a) states that if µ âMF (Rd) satisfies (F1)-s and β â¤ Î²â(Î±, s) = Î±(dâs)+Î± ,then µ(Xt) > 0 almost surely on {Xt 6= 0} and, equivalently, uâµt = Ut.Without loss of generality, we can assume that supp(µ) is bounded. Indeed, if (F1)-sholds for µ, then it also holds for the restriction of µ to a bounded set. Furthermore, if µâ²denotes this restriction, then uÎ»µâ²t â¤ uλµt for λ > 0 by the comparison principle, and henceit suffices to show that uâµâ²t = Ut. We will further assume that supp(µ) â B1 = {x â Rd :|x| â¤ 1}. This is allowable because, by translation invariance of (4.1.3), uâµt is flat if andonly if uâµzt is flat, where µz is the translate of µ by z â Rd. We can therefore translateµ so that it has positive mass in B1, then discard the mass outside B1 by the previousargument.We set the following standing assumption: for the remainder of this section, let µ âMF (Rd) satisfy (F1)-s for some s â [0, d] and µ(Bc1) = 0. Without loss of generality wewill suppose that µ(B1) = µ(Rd) = 1.The proof of Theorem 4.1.8(a) uses a similar argument to the proof of Proposition 4.3.1.Recall that the bound uâ(1, x) â¥ cp1(x) from Lemma 4.3.2 played a critical role in thatresult. We use a similar bound here, which however is adapted to uâµt for µ satisfying(F1)-s. By monotonicity of λ â uλµt , we have the trivial bound that uâµt â¥ uλµt for allλ > 0. As we will also be using scaling properties of these solutions, which involve rescalingthe initial measure (see (4.2.23)), the critical scale turns out to be ursµ(Â·/r)(1, x), where werecall that for r > 0, µ(Â·/r) is the measure given by µ(A/r) = â« 1A(rx)dµ(x). If µ hassupport S, then the support of µ(Â·/r) is rS = {rx : x â S}. The next result, which isanalogous to Lemma 4.3.2, gives a lower bound for ursµ(Â·/r)(1, x).193Lemma 4.5.1. Let µ â MF (Rd) satisfy (F1)-s for some s â [0, d]. Then there is a constantc4.5.1 = c4.5.1(µ, α, β, d) > 0 such that for all r â¥ 1,ursµ(Â·/r)(1, x) â¥ c4.5.1 rsS1(µ(Â·/r))(x).The proof of Lemma 4.5.1 requires the following boundedness result.Lemma 4.5.2. Let µ â MF (Rd) satisfy (F1)-s for s â [0, d] and fix α â (0, 2). Then thereis a constant C4.5.2 = C4.5.2(µ, α, d) > 0 such that for all t > 0,suprâ¥1supyâRdSt(rsµ(Â·/r))(y) â¤ C4.5.2 tâ(dâs)Î± .Proof. Fix t, r > 0. For y â Rd, we haveSt(rsµ(Â·/r))(y) = rsâ«pt(zË â y)dµ(zË/r)= rsâ«pt(rz â y)dµ(z)= rsâ« â0µ ({z : pt(rz â y) â¥ k}) dk,where the last line uses Fubiniâs theorem. By (4.2.2), it follows that{z : pt(rz â y) â¥ k} â {z : C4.2.1t|rz â y|d+α â¥ k} â© {C4.2.1tâ dα â¥ k}.Note that the first set on the right hand side is equal to B(y/r, c(kâ1t)1d+Î± râ1) with c =C1d+α4.2.1. Using this and the fact that µ satisfies (F1)-s, we haveSt(rsµ(Â·/r))(y) â¤ rsâ« C4.2.1tâ dα0µ(B(y/r, c(kâ1t)1d+Î± râ1))dkâ¤ Crsrâst sd+Î±â« C4.2.1tâ dÎ±0kâsd+Î±dkâ¤ Ctâ (dâs)Î± ,where we recall that s â¤ d and so C depends only on C, d and α.Proof of Lemma 4.5.1. As in the proof of Lemma 4.3.2, we will use the Feynman-Kac for-mula from Section 4.2.4. We cannot apply Lemma 4.2.11 directly to ââÎ»uÎ»rsµ(Â·/r) because194rsµ(Â·/r) is not a function, so for Ç« > 0 we defineÏÇ«,r = Sǫ(rsµ(·/r)) = rs(µ(Â·/r) â pǫ). (4.5.1)Then ÏÇ«,r is a smooth, bounded, non-negative function, so by Lemma 4.2.11(b), wÎ»ÏÇ«,r (t, x) =ââÎ»uÎ»ÏÇ«,r(t, x) exists andwÎ»ÏÇ«,r (t, x) = EWx(ÏÇ«,r(Wt) exp(â(1 + Î²)â« t0uÎ»ÏÇ«,r (tâ Ï,WÏ )Î²dÏ)). (4.5.2)By (4.2.16), we haveuÎ»ÏÇ«,rÏ â¤ SÏ (Î»ÏÇ«,r) = Î»SÏ (ÏÇ«,r)for Ï > 0. Hence by (4.5.2) with t = 1 we havewÎ»ÏÇ«,r(1, x) â¥ EWx(ÏÇ«,r(W1) exp(â(1 + Î²)Î»Î²â« 10[S1âÏ (ÏÇ«,r)(WÏ )]Î²dÏ))= EWx(ÏÇ«,r(W1) exp(â(1 + Î²)Î»Î²â« 10[S1âÏ+Ç«(rsµ(Â·/r))(WÏ )]Î²dÏ)),where to obtain the final expression we have used (4.5.1) and the semigroup property. ByLemma 4.5.2,[S1âÏ+Ç«(rsµ(Â·/r))(WÏ )]β â¤ CÎ²4.5.2(1â Ï + Ç«)â (dâs)Î²Î± ,and in particular there is a constant C > 0 such that for 0 < Ç« â¤ 1,[S1âÏ+Ç«(rsµ(Â·/r))(WÏ )]β â¤ C(1â Ï + Ç«)âdÎ²Î± .Using this bound and changing variables, we obtain thatwÎ»ÏÇ«,r(1, x) â¥ EWx (ÏÇ«,r(W1)) exp(âC(1 + Î²)Î»Î²â« 10(u+ Ç«)âdÎ²Î± du).Since β < αd the integral remains bounded as Ç« â 0. Hence for a new constant C > 0, weobtain that for all 0 < Ç« â¤ 1,wÎ»ÏÇ«,r (1, x) â¥ EWx (ÏÇ«,r(W1)) exp(âCÎ»Î²). (4.5.3)We now integrate over λ to obtain a lower bound for uÏÇ«,r(1, x). Since wÎ»ÏÇ«,r(1, x) =195ââÎ»uÎ»ÏÇ«,r(1, x), by (4.2.29) and (4.5.3) we haveuÏÇ«,r(1, x) =â« 10wÎ»ÏÇ«,r (1, x) dÎ»â¥ EWx (ÏÇ«,r(W1))â« 10exp(âCÎ»Î²)dÎ»â¥ c0EWx (ÏÇ«,r(W1)) (4.5.4)for a constant c0 > 0. It remains to show that the left and right hand sides of the aboveconverge to the desired quantities when Ç« â 0. It follows from Lemma 4.2.6 thatlimÇ«â0uÏÇ«,r(1, x) = ursµ(Â·/r)(1, x). (4.5.5)Turning to the right hand side of (4.5.4), we first observe that EWx (ÏÇ«,r(W1)) = S1ÏÇ«,r(x).Thus we may use the Dominated Convergence Theorem to see thatlimÇ«â0EWx (ÏÇ«,r) = S1(rsµ(Â·/r))(x). (4.5.6)Letting Ç« â 0 in (4.5.4), from (4.5.5) and (4.5.6) we obtain thatursµ(Â·/r)(1, x) â¥ c0 rsS1(µ(Â·/r))(x)for all x â Rd, which completes the proof.We now have all the tools we need to prove Theorem 4.1.8(a). Part (b) is proved inSection 4.7.Proof of Theorem 4.1.8(a). Fix X0 â MF (Rd). Let µ â MF (Rd) satisfy (F1)-s as well asour assumptions that µ(1) = 1 and µ(Bc1) = 0. By Lemma 4.2.9, in particular using (4.2.20)with uλµt and taking Î»ââ, we obtainPXX0(µ(Xt) = 0) = exp(âX0(uâµt )). (4.5.7)Let δn = 2ân. Assume that n is large enough so that δn < t, and consider the conditionalprobability P (µ(Xt) = 0 | FtâÎ´n). Applying the Markov property and using (4.5.7), we196obtain thatP (µ(Xt) = 0 | FtâÎ´n ) = PXXtâÎ´n (µ(Xδn) = 0)= exp(ââ«uâµ(Î´n, x)dXtâÎ´n(x))â¤ exp(ââ«B2uâµ(Î´n, x)dXtâÎ´n (x)), (4.5.8)where B2 = {x â Rd : |x| â¤ 2}. Using monotonicity of λ â uλµ(δn, x) and the scalingrelationship (4.2.24), we haveuâµ(Î´n, x) â¥ uÎ´â sÎ±âÎ±âÎ²Î±Î²n µ(δn, x)= Î´â 1βn uÎ´â sαn µ(Â·/Î´â1Î±n )(1, Î´â 1αn x)â¥ c4.5.1 Î´â 1Î²â sαn S1(µ(Â·/Î´â1Î±n ))(Î´â 1αn x). (4.5.9)The final inequality follows from Lemma 4.5.1. We expand the semigroup term in the aboveas a convolution with p1. After a change of variables, we haveS1(µ(Â·/Î´â1Î±n ))(Î´â 1αn x) =â«p1(Î´â 1αn (xâ y)) dµ(y)â¥ µ(1) p1(Î´â1Î±n d(x,S)), (4.5.10)where S = supp(µ) and d(x,S) = infyâS |xâ y|, and we recall that for Ï > 0, p1(Ï) denotesp1(|z|) with |z| = Ï. Because µ is supported on B1, for any x â B2 we have d(x,S) â¤ 3. Inparticular, using this in (4.5.10) and substituting it into (4.5.9), we obtainuâµ(Î´n, x) â¥ c4.5.1 Î´â 1Î²â sαn p1(3Î´â 1αn ) for all x â B2,where µ(1) does not appear because it equals one. Using (4.2.2) to bound p1 below, weconclude that for a constant c1 > 0,uâµ(Î´n, x) â¥ c1Î´â 1Î²â sαn (Î´â 1αn )â(d+Î±)= c1Î´âqn for all x â B2, (4.5.11)where q := 1β â dâs+Î±Î± . Using (4.5.11) in (4.5.8), we obtain the following:P (µ(Xt) = 0 | FtâÎ´n ) â¤ exp(âc1XtâÎ´n(B2) Î´âqn ) . (4.5.12)197From this point, the proof is identical to that of Proposition 4.3.1. By instantaneouspropagation, Xt(B2) > 0 almost surely on {Xt 6= 0}. One considers the event A2κ ={Xt(B2) â¥ 2κ} for κ > 0 and notes that XtâÎ´n(B2) â¥ κ eventually a.s. on A2κ. This leadsto a statement analogous to (4.3.9). One then finishes the proof in the same way: by directcomputation when β < Î²â(Î±, s) and using martingale convergence when β = Î²â(Î±, s). Thiscompletes the proof that PXX0(µ(Xt) = 0 | Xt > 0). The result under Nx follows from (4.2.32),which implies thatNx(µ(Xt) = 0 | Xt 6= 0) = 0.In particular, we have Nx(µ(Xt) > 0) = Nx(Xt 6= 0) = Ut. Since uâµt (x) = Nx(µ(Xt) > 0),this proves the last claim.4.6 Decay of Nx(µ(Xt) > 0) for (F2)-s measures whenβ > Î²â(Î±, s)This section is concerned with establishing conditions under which Nx(µ(Xt) > 0) = uâµt (x)is non-flat (and hence µ(Xt) = 0 with positive probability on {Xt 6= 0}) and quantifying itsasymptotic behaviour under these conditions. The main result we prove is Theorem 4.1.10.The proofs are analytic and we pose it as an open problem to prove the same results usingprobabilistic arguments.We will show that uâµt = Nx(µ(Xt) > 0) is non-flat when β > Î²â(Î±, s) = Î±dâs+Î± , wheres â [0, d] and µ has compact support and satisfies (F2)-s, which we recall is the conditionthat(F2)-s For some constant C > 0, for all x â supp(µ) and r â (0, 1],µ(B(x, r)) â¥ Crs.The method we use is to show the existence certain barrier functions for the equation(4.1.3). A function h : Qâ R+ is a barrier function for µ if it is a super-solution to (4.1.3)on Q that explodes on supp(µ) with order tâ1Î² and vanishes on supp(µ)c as t â 0. Ourmethod is based on, and adapted from, a similar argument in [7].First, we define W : R+ â R+ byW (r) =log(e+ r2)1 + rd+α. (4.6.1)198We also introduce V : Rd â R+, given byV (x) =W (|x|) = log(e+ |x|2)1 + |x|d+α . (4.6.2)For (t, x) â Q, we then define wt(x) bywt(x) = tâ 1β (1 + tâsÎ± )V (tâ1Î±x). (4.6.3)Finally, for k > 0 and µ â MF (Rd), let hk(t, x) be given byhk(t, x) = k(wt â µ)(x). (4.6.4)Note that wt â C1,2(Q), the space of functions which are once continuously differentiablein time and twice continuously differentiable in space. Consequently, we also have thathk â C1,2(Q). Recall that Î²â(Î±, s) = Î±(dâs)+Î± . In what follows, we restrict to s â [0, α),since this is required to have Î²â(Î±, s) < αd .For closed S â Rd, recall that MF (S) is the space of measures µ â MF (Rd) withsupp(µ) â S, and that d(x,S) = infyâS |xâ y|.Proposition 4.6.1. Suppose that µ â MF (Rd) satisfies (F2)-s for some s â [0, α) and hascompact support S â Rd. Let Î²â(Î±, s) < β < αd .(a) There exists Î0 > 0 such that if k â¥ Î0, hk is a (strong) supersolution to (4.1.3) on Q,in the sense that for all (t, x) â Q,(ât ââÎ±)hk(t, x) + hk(t, x)1+β â¥ 0. (4.6.5)(b) For x â S and t > 0,hk(t, x) â¥ c4.6.6ktâ1Î² , (4.6.6)where c4.6.6 = C · c0 and c0 > 0 depends only on (α, d). For all (t, x) â Q,hk(t, x) â¤ kµ(1)[tâ1Î²â sα â¨ tâ 1β ]W (tâ 1αd(x,S)). (4.6.7)In particular, limtâ0 hk(t, x) = â for x â S, and hk(t, Â·) vanishes uniformly on {x :d(x,S) â¥ Ï} as t â 0 for all Ï > 0.(c) For (t, x) â Q we havehk(t, x) â¥ kc4.6.8tâ1Î²W (tâ1Î±d(x,S)), (4.6.8)where c4.6.8 = C · c1 and c1 > 0 depends only on (α, d).199(d) For any ν â MF (S), if k â¥ Î0 then hk(t, x) â¥ uν(t, x) on Q.This proposition is the main result underlying Theorem 4.1.10 (as well as part (b) ofTheorem 4.1.14). Before proving it, we comment on the technique. By and large, ourmethod is adapted from the argument used by Chen, Veron and Wang in [7] to prove theresult we called Theorem 4.B(b) in the introduction. Our barrier function is modelled aftertheirs and we make use of some of their intermediate results. Define wËt bywËt(x) = tâ 1βW (tâ1Î± |x|).In [7], it is shown that kwËt(x) is a supersolution to (4.1.3) for sufficiently large k. Thisbarrier function is what the authors use to prove that limÎ»ââ uλδ0(t, x) is non-flat whenβ > αd+α . Part of their proof was a detailed analysis of ââÎ±V . In equation (5.11) of [7],the following bound is established: there is a constant c1 > 0 such that for x â Rd with|x| â¥ 2,ââÎ±V (x) â¥ â c11 + |x|d+α . (4.6.9)This bound is critical to their argument and it is equally critical in ours which follows. Ascan be seen from (4.6.4), the function hk(t, x) is essentially wËt spread out over S = supp(µ)via a convolution with µ, with an additional power of t to locally normalize mass of µ whent â¤ 1. By spreading out wËt over S, we construct a supersolution which is singular on S ast â 0.We make a few observations about the functions we have introduced. The functionW (z)is not globally decreasing for positive z, and correspondingly V and wt are not globallyradially decreasing. However, for any d â¥ 1 and α â (0, 2), W attains its maximum valueat some r0 â [0, 1) and W (r) is decreasing for r â¥ r0. Furthermore, one can verify thatminrâ[0,1]W (r) = W (1), and so for all d â¥ 1 and α â (0, 2), W is weakly decreasing in thesense thatminrâ²â[0,r]W (râ²) =W (r) for all r â¥ 1. (4.6.10)V inherits this as a form of weak radial decreasing, i.e.minxâB(0,R)V (x) =W (R) for all R â¥ 1. (4.6.11)Finally, one can show that there is a constant c4.6.12 > 0 such thatFor all R â¥ 0, sup|x|â¥RV (x) â¤ c4.6.12W (R). (4.6.12)Proof of Proposition 4.6.1(a). First let us consider the time derivative of wt(x). Expanding200directly using (4.6.2) and (4.6.3), for z = tâ1Î± |x| we haveât(wt(x)) = â 1Î²tâ 1Î²â1W (tâ1Î± |x|) +(â 1Î²â sÎ±)tâ 1Î²â sÎ±â1W (tâ1Î± |x|)â 1Î±tâ1Î²â1(1 + tâsÎ± )(t1α |x|)W â²(tâ 1α |x|)= tâ 1Î²â sÎ±â1[(â 1Î²â sα)W (z)â 1αzW â²(z)]+ tâ1Î²â1[â 1βW (z)â 1αzW â²(z)]. (4.6.13)Computing W â² directly, we obtain thatW â²(z) =2ze+ z211 + zd+Î±â (d+ α) log(e+ z2)zd+Î±â1(1 + zd+α)2From (4.6.13), it follows thatât(wt(x)) = tâ 1Î²â sÎ±â1W (z)[(â 1Î²â sÎ±)â 1α2z2(e+ z2)â1log(e+ z2)+d+ ααzd+α1 + zd+α]+ tâ 1Î²â1W (z)[â 1Î²â 1α2z2(e+ z2)â1log(e+ z2)+d+ ααzd+α1 + zd+α]= tâ1Î²â sÎ±â1W (z)f1(z) + tâ 1Î²â1W (z)f2(z). (4.6.14)Consider f1(z), the square bracketed quantity in the first line. The second term in f1(z)vanishes as z ââ, and the third converges to d+αα . The assumption β > Î±dâs+Î± is equivalentto 1β +sα <d+αα ; it follows that f1(z) is positive for sufficiently large z. Moreover, there areconstants R0 > 0 and c2 > 0 such that the for z â¥ R0, f1(z) â¥ c2. Since f2(z) > f1(z) forall z â¥ 0, we also have f2(z) â¥ c2 for z â¥ R0, and from (4.6.14) we have the following: forall (t, x) satisfying tâ1Î± |x| â¥ R0,ât(wt(x)) â¥ c2tâ1Î²â1(1 + tâsÎ± )W (tâ1Î± |x|). (4.6.15)Next we consider âÎ±wt. By the scaling of the α-stable process, if gλ(x) = g(λx), thenâÎ±gÎ»(x) = Î»Î±(âÎ±g)(Î»x). Using this and (4.6.3), it follows thatâÎ±wt(x) = tâ 1Î²â1(1 + tâsÎ± )(âÎ±V )(tâ 1αx). (4.6.16)Using the above and (4.6.9), for all (t, x) such that tâ1Î± |x| â¥ 2,ââÎ±wt(x) â¥ âc1tâ1Î²â1(1 + tâsÎ± )11 + |tâ 1αx|d+α. (4.6.17)201This bound allows a direct comparison with ât(wt(x)). In particular, by (4.6.15) and(4.6.17), (and recalling the definition of W from (4.6.1)) we have, for |tâ 1αx| â¥ 2 â¨R0,(ât ââÎ±)wt(x) â¥ tâ 1Î²â1(1 + tâsÎ± )1 + |tâ 1αx|d+α[c2 log(e+ |tâ 1Î±x|2)â c1].It follows that for some R1 â¥ R0 â¨ 2,(ât ââÎ±)wt(x) â¥ 0 for all (t, x) satisfying |tâ1Î±x| â¥ R1. (4.6.18)Now consider hk(t, x). We can take differentiation under the integral in (4.6.4) to obtain(ât ââÎ±)hk(t, x) = kâ«(ât ââÎ±)wt(xâ y)dµ(y). (4.6.19)Recall that d(x,S) denotes the distance from x â Rd to the set S. If tâ 1αd(x,S) â¥ R1, from(4.6.18) the integrand in (4.6.19) is positive for all y â S, i.e. for all y â supp(µ), and hence(ât ââÎ±)hk(t, x) â¥ 0 for all (t, x) satisfying tâ1Î±d(x,S) â¥ R1. (4.6.20)The condition on (t, x) that tâ1Î±d(x,S) â¥ R1 will be important, so we introduceQâ¥R1 = {(t, x) â Q : tâ 1α d(x,S) â¥ R1}.The statement (4.6.20) then reads that(ât ââÎ±)hk(t, x) â¥ 0 for all (t, x) â Qâ¥R1 . (4.6.21)We also introduceQ<R1 = {(t, x) â Q : tâ 1α d(x,S) < R1},and now consider the behaviour of (ât â âÎ±)hk on Q<R1 . We can apply (4.6.18) to theintegrand in (4.6.19) to obtain that(ât ââÎ±)hk(t, x) â¥ kâ«B(0,t1Î±R1)(ât ââÎ±)wt(y)dµ(y â x). (4.6.22)202By (4.6.13) and (4.6.16), for u = tâ1Î±x we have(ât ââÎ±)wt(y) = tâ1Î²â sÎ±â1[(â 1Î²â sα)W (|u|)â 1α|u|W â²(|u|)ââÎ±V (u)],+ tâ1Î²â1[â 1βW (|u|)â 1α|u|W â²(|u|)ââÎ±V (u)].Since all the terms in above are continuous functions of u and |u| â¤ R1 on {y : |y| â¤ t 1αR1},the square-bracketed terms above are bounded on this set. Thussupy:|y|â¤t 1Î±R1|(ât ââÎ±wt(y)| â¤ c3tâ1Î²â1[tâsÎ± â¨ 1]for some 0 < c3 <â. Using this in (4.6.22) we obtain that(ât ââÎ±)hk(t, x) â¥ âkc3tâ1Î²â1[µ(B(x, t1αR1))tsα â§ 1](4.6.23)for all (t, x) â Q. We now must show that the non-linear term in (4.1.3), given byhk(t, x)1+β , is sufficiently large on Q<R1 so that hk(t, x) is a super-solution to (4.1.3) evenin the case of the worst-case bound given in (4.6.23). From (4.6.3) and (4.6.4), we have1k1+βhk(t, x)1+β =[â«wt(xâ y)dµ(y)]1+Î²=[tâ 1β (1 + tâsÎ± )â«V (tâ1Î± (xâ y))dµ(y)]1+Î²â¥[(tâ1Î²â sα â¨ tâ 1β )â«B(x,t1Î± (R1+1))V (tâ1Î± (xâ y))dµ(y)]1+βwhere the final inequality holds because V â¥ 0. Restricted toB(x, t 1α (R1+1)), V (tâ 1α (xây))is bounded below by W (R1 + 1) by (4.6.11). This implies that1k1+βhk(t, x)1+β â¥ c4tâ1Î²â1[µ(B(x, t1α (R1 + 1)))tsα â§ 1]1+β(4.6.24)on Q<R1 , where c4 = W (R + 1)1+β . Combining (4.6.23) and (4.6.24), we obtain that for203x â Q<R1T ,(ât ââÎ±)hk(t, x) + hk(t, x)1+Î²â¥ ktâ 1Î²â1ï£«ï£­c4kÎ²[µ(B(x, t1α (R1 + 1)))tsα â§ 1]1+Î²â c3[µ(B(x, t1αR1))tsα â§ 1]ï£¶ï£¸â¥ ktâ 1Î²â1[µ(B(x, t1αR1))tsα â§ 1]ï£«ï£­c4kÎ²[µ(B(x, t1α (R1 + 1)))tsα â§ 1]Î²â c3ï£¶ï£¸ . (4.6.25)Since (t, x) â Q<R1 , there must be a point y0 â S such that B(y0, t 1α ) â B(x, t 1α (R1 + 1)).In particular, since µ satisfies (F2)-s we haveµ(B(x, t1α (R1 + 1))) â¥ µ(B(y0, t1α )) â¥ C[t sα â§ 1].The minimum above appears since for y0 â S and t â¥ 1, µ(B(y0, t 1α )) â¥ µ(B(y0, 1)) â¥ C.Using this in (4.6.25), we obtain(ât ââÎ±)hk(t, x)â hk(t, x)1+β â¥ ktâ1Î²â1[µ(B(x, t1αR1))tsα â§ 1](câ²4kÎ² â c3)for all (t, x) â Q<R1 , where câ²4 = Cβc4. For sufficiently large k the above is non-negative,and hence hk(t, x) is a super-solution to (4.1.3) on Q<R1 . On the other hand, becausehk(t, x)1+β â¥ 0, (4.6.21) implies that hk(t, x) is a super-solution on Qâ¥R1 . Thus we haveshown that for sufficiently large k, hk(t, x) is a super-solution to (4.1.3) on Q<R1 âªQâ¥R1 =Q.Proof of Proposition 4.6.1(b)-(d). We first show part (c). For fixed x, since S is closed,there is a point y0 â S such that |xâ y0| = d(x,S). Hence from (4.6.4),hk(t, x) â¥ kâ«B(y0,t1Î± )wt(xâ y)dµ(y)= ktâ1Î² (1 + tâsÎ± )â«B(y0,t1Î± )W (tâ1Î± |xâ y|)dµ(y)â¥ k[tâ 1β â¨ tâ 1Î²â sα ]W (tâ 1αd(x,S) + 1)µ(B(y0, t 1α )),where the last line has used the triangle inequality and (4.6.10). Since µ satisfies (F2)-s,µ(B(y0, t1α )) â¥ Ct sα for t â¤ 1 and µ(B(y0, t 1α )) â¥ C for t > 1. This implies that above isbounded below by kCtâ 1βW (tâ1Î± d(x,S) + 1). The result then follows from (4.6.12).Next we prove part (b). The claim for x â S follows from part (c). Now let x â Rd.204Applying (4.6.12), we obtain that for every y â S,wt(xâ y) â¤ c4.6.12[tâ1Î²â sα â¨ tâ 1β ]W (tâ 1αd(x,S)).One then uses this bound directly in the convolution defining hk(t, x), i.e. (4.6.4), to obtain(4.6.7). Using formula (4.6.1) forW and the fact that 1β+sα <d+αα , the uniform convergenceas t â 0 follows.We now prove (d). Suppose that ν â MF (S). (This includes the case ν = µ.) Inorder to show that uν(t, x) â¤ hk(t, x), we will need to consider a sequence of solutionscorresponding to a sequence of approximations of ν. Let Z > 0 be the normalizing constantsuch thatZâ1â«V (x)dx = 1.Let VË (Â·) = Zâ1V (Â·) and define a sequence of C2 mollifiers by Ïn(x) = ndVË (nx) for n â¥ 1.Then it is immediate that ν â Ïn â ν in the weak sense of measures as n â â. Letun(t, x) = uÎ½âÏn(t, x). By Lemma 4.2.6(a),limnââun(t, x) = uν(t, x) (4.6.26)for all (t, x) â Q. For x â Rd,ν â Ïn(x) = 1Zndâ«W (n|y â x|)dÎ½(y)â¤ 1Zν(1)nd supyâSW (n|xâ y|)Hence from (4.6.12) we obtain thatν â Ïn(x) â¤ c4.6.12ν(1)Z[ndW (nd(x,S))]. (4.6.27)Let tn = nâÎ±. By (4.6.8), we havehk(tn, x) â¥ c4.6.8knαβW (nd(x,S))= c4.6.8knÎ±Î²âd[ndW (nd(x,S))] (4.6.28)By part (a), if k â¥ Î0, then (t, x) â hk(tn + t, x) is a super-solution to (4.1.3) on Q withinitial value hk(tn, Â·). Since αβ > d, it follows from (4.6.27) and (4.6.28) that, for sufficientlylarge n,ν â Ïn(x) â¤ hk(tn, x).205Since un has initial data ν â Ïn and (t, x)â hk(tn + t, x) is a super-solution, the above andthe comparison principle imply thatun(t, x) â¤ hk(tn + t, x)for all (t, x) â Q. Taking nââ on both sides and using (4.6.26), we obtain thatuν(t, x) â¤ hk(t, x)for all (t, x) â Q.The final ingredient in the proof of Theorem 4.1.10 is the following lower bound onuâµt (x). We recall that uât (x) = limÎ»ââ uλδ0t (x).Lemma 4.6.2. Let µ â MF (Rd). Then for every z â supp(µ),uâµt (x) â¥ uât (xâ z) â¥ c4.6.2tâ1Î² p1(tâ 1α (xâ z))for all (t, x) â Q, where c4.6.2 > 0 depends only on (α, β, d).We defer the proof to the end of the section.Proof of Theorem 4.1.10. Fix s â [0, α) and let µ satisfy (F2)-s and have compact supportS. Let Î²â(Î±, s) < β < αd . We begin with part (b). We need to show that the upper boundin (4.1.19) holds, i.e. for some constant C > 0,uâµt (x) â¤ C[tâ1Î²â sα â¨ tâ 1β ]W (tâ 1α d(x,S))on Q. Let Î0 > 0 be as in Proposition 4.6.1(a). Then by Proposition 4.6.1(d), uλµt (x) â¤hÎ0(t, x) for every λ > 0, and hence uâµt (x) = limÎ»ââ uλµt (x) â¤ hÎ0(t, x). The abovebound then follows from Proposition 4.6.1(b) with C = µ(1)Î0, which proves the upperbound in (4.1.19). The lower bound in (4.1.19) follows from Lemma 4.6.2.Next we prove part (a). From the upper bound in (4.1.19), it is clear that for t > 0,uâµt (x) â 0 as d(x,S) â â. We can thus take x0 â Rd so that uâµt (x0) < Ut, which isequivalent to Nx0(µ(Xt) = 0 | Xt 6= 0) > 0. By (4.2.31), we have PXδx0 (µ(Xt) = 0 | Xt 6= 0) >0. Appealing to (4.2.5), it follows thatPXX0(µ(Xt) = 0 | Xt 6= 0) > 0for every X0 â MF (Rd). In particular, we can take X0 = δx and use (4.2.31) again to seethat Nx(µ(Xt) = 0 | Xt 6= 0) > 0 for every x â Rd. This completes the proof of (a).206To see that part (d) holds, recall that Proposition 4.6.1(d) applies to any ν â MF (S).The upper bound in (4.1.21) then follows by the same argument used to prove the upperbound in (4.1.19) above. The argument used to prove (a) can then be used to prove thatthe claims from (a) hold when ν is replaced with µ. Finally, as in the proof of part (b), thelower bound in (4.1.21) follows from Lemma 4.6.2.It remains to show part (c). Recall from (4.2.25) thatuâµt (x) = tâ 1β uâµ(Â·/tâ 1α )1 (tâ 1αx).If µ satisfies (F1)-s, for t â¤ 1 we apply Lemma 4.5.1 to the right hand side of the above toobtainuâµt (x) â¥ c4.5.1tâ1Î²â sÎ±â«p1(y â tâ 1Î±x)dµ(y/tâ1Î±)= c4.5.1tâ 1Î²â sÎ±â«p1(tâ 1α (y â x))dµ(y).For every y â S, |y â x| â¤ d(x,S) + diam(S), so from the above, for x â Rd and t â¤ 1 wehaveuâµt (x) â¥ c4.5.1µ(1)tâ1Î²â sα p1(tâ 1α (d(x,S) + diam(S))).The lower bound on p1 from (4.2.1) then implies (4.1.20).Proof of Lemma 4.6.2. Let µ â MF (Rd) and z â supp(µ). Then µ(B(z, Ï)) > 0 for everyÏ > 0. For k > 0 and Ï > 0, we havelim inftâ0â«B(z,Ï)ukµt (x) dx â¥ kµ(B(z, Ï/2)).Since uâµt â¥ ukµt for all k > 0, it follows thatlimtâ0â«B(z,Ï)uâµt (x) dx = +â for all z â supp(µ) and Ï > 0. (4.6.29)We now fix λ > 0 and z â supp(µ). By (4.6.29), there exists (tn, Ïn) such that tn, Ïn â 0and â«B(z,Ïn)uâµtn (x) dx = λfor all n â¥ 1. Let Ïn(x) = uâµtn (x)1B(z,Ïn)(x). Note that Ïn â λδ0(· â z) in the weak senseof measures as nââ, so by Lemma 4.2.6(a), limnââ uÏnt (x) = uλt (xâ z). By (4.1.10), wehaveuÏnt (x) = Nx(1â exp(Xt(Ïn))). (4.6.30)207On the other hand, we remark that for every k > 0, by (4.2.19) and the Markov property,ukµt+tn(x) = Nx(1â exp(kµ(Xt+tn))= Nx(1â exp(Xt(ukµtn )).Taking k ââ, we obtainuâµt+tn(x) = Nx(1â exp(Xt(uâµtn )). (4.6.31)By the definition of Ïn, we have Ïn â¤ uâµtn for all n â N. Hence, from (4.6.30) and (4.6.31),we haveuÏnt (x) â¤ uâµt+tn(x)for all (t, x) â Q and all n â N. As noted above, the left hand side converges to uλt (x â z)as nââ. Taking nââ, we obtain thatuλt (xâ z) â¤ uâµt (x).Taking λ â â, we obtain uât (x â z) â¤ uâµt (x). This proves the first inequality in thelemma, and the second then follows from (4.3.5).4.7 The initial trace problemWe now apply the results of Sections 4.5 and 4.6 to the initial trace theory for (4.1.3). Recallthat the initial trace of a solution ut(x) to (4.1.3) was defined in (4.1.23) and (4.1.24). Werestate the definition here for convenience. A pair (S, ν) with S â Rd a closed set and ν aRadon measure with ν(S) = 0 is the initial trace of ut(x) if:â¢ For all ξ â Cc(Sc).limtâ0â«Î¾(x)ut(x)dx =â«Î¾dÎ½. (4.7.1)â¢ For every z â S and Ï > 0,limtâ0â«B(z,Ï)ut(x)dx = +â. (4.7.2)Our contribution is to the problem of determining when a solution with a given initial traceexists. We consider weak solutions. First, recall that we have defined a weak solution to(4.1.3) in Definition 4.2.3. Although we only consider the problem for initial traces in whichthe regular component (i.e. the Radon measure) is null, our definition applies for general208initial traces.Definition 4.7.1. For closed S â Rd and a Radon measure ν satisfying ν(S) = 0, we saythat u : Q â [0,â) is a weak solution to the initial trace problem with initial trace (S, Î½)if:â¢ u is a weak solution to (4.1.3) in the sense of Definition 4.2.3.â¢ The initial trace (S, ν) is attained in the sense that (4.7.1) and (4.7.2) hold.Our main result about the initial trace problem is Theorem 4.1.14, which has two parts:non-existence and existence. Most of the work has already been carried out in Sections 4.5and 4.6. Both proofs require the following lemma.Lemma 4.7.2. For µ â MF (Rd), uâµt (x) = limÎ»ââ uλµt (x) is a weak solution to (4.1.3).Furthermore, for all Ï > 0 and z â supp(µ),limtâ0â«B(z,Ï)uâµt (x) dx = +â, (4.7.3)and hence the singular set of ut contains supp(µ).Proof. Let µ â MF (Rd). We must show that uâµ(t, x) satisfies (4.2.13) for every ξ âC1,2c (Q). For λ > 0, uλµ(t, x) is a solution to the problem (4.2.14) with u0 = λµ, so bydefinition we haveâ«Q(uÎ»µ(t, x)[ââtÎ¾(t, x)ââÎ±Î¾(t, x)]) + uλµ(t, x)1+βξ(t, x) dxdt = 0.Since ξ has compact support, by (4.1.18) the bound uλµ(t, x) â¤ uâµ(t, x) â¤ Ut allowsus to apply Dominated Convergence and conclude that uâµ(t, x) satisfies (4.2.13) for allξ â C1,2c (Q). Similarly, uâµ(t, x) â¤ Ut implies that uâµ(t, x) is bounded on (Ç«,â) Ã Rdfor all Ç« > 0 and hence uâµ â L1+βloc (Q). To see the uâµt (x) is continuous, we note that forany t0 > 0, (t, x) â uâµ(t0 + t, x) is a weak solution to (4.1.3) which is globally boundedby Ut0 . In particular, (recall Remark 4.2.8) it is a solution to the integral equation (4.1.2)with initial data uâµt0 â B+b and hence is continuous. It follows that uâµt (x) is continuouson [t0,â)Ã Rd for all t0 > 0 and hence on Q. Thus uâµt (x) is a weak solution to (4.1.3).The fact that (4.7.3) holds has already been shown in the proof of Lemma 4.6.2; see(4.6.29).Our main result concerning flatness and non-existence is the following. We recall from(4.1.26) the space U of positive functions on Q bounded above by Ut, and that we haverestricted our attention to solutions in U .209Theorem 4.7.3. Suppose that S â Rd is closed and supports a measure µ â MF (Rd) forwhich limÎ»ââ uλµt = Ut. If S is contained in the singular set of a solution u to (4.1.3) inU , then ut = Ut. In particular, if S 6= Rd there is no solution to (4.1.3) in U with singularset S.We obtain Theorem 4.1.8(b) and Theorem 4.1.14(a) as corollaries as follows.Proof of Theorem 4.1.8(b). Let s â [0, d], β â¤ Î²â(Î±, s) and suppose that µ â MF (Rd)satisfies (F1)-s. Then uâµt (x) = Ut by Theorem 4.1.8(a). Hence S = supp(µ) supports ameasure µ for which limÎ»ââ uλµt (x) = Ut. Now let ν âMF (Rd) be any measure such thatsupp(µ) â supp(ν). By Lemma 4.7.2, uâÎ½t is a weak solution to (4.1.3) whose singular setcontains supp(ν), which itself contains S. The result follows from Theorem 4.7.3.Proof of Theorem 4.1.14(a). If S â Rd is closed with Hdsat(S) > 0, then by FrostmanâsLemma there exists µ â MF (S) which satisfies (F1)-dsat. By Theorem 4.1.8(a), uâµt = Ut,and the result follows from Theorem 4.7.3.The proof of Theorem 4.7.3 requires the following pointwise estimate for solutions witha given singular set. A more general version of this result proved for classical solutions in[8], where it was called Theorem C. We include a short proof for the sake of completeness.The argument is essentially the same as the argument in the proof of Lemma 4.6.2. Recallthat uât (x) = limÎ»ââ uλδ0t (x).Proposition 4.7.4. Suppose that u(t, x) is a weak solution to (4.1.3) in U whose singularset contains S â Rd. Then for every z â S,u(t, x) â¥ uâ(t, xâ z) â¥ c4.7.4tâ1Î² p1(tâ 1α (xâ z))for all (t, x) â Q, where c4.7.4 > 0 depends only on (α, β, d).Proof. Suppose that z â S, the singular set of u(t, x). Let λ > 0. Since (4.7.2) holds, theremust be sequences (tn)nâ¥1 and (Ïn)nâ¥1 such that tn, Ïn â 0 andâ«B(z,Ïn)u(tn, x) dx = λ.Let Ïn(x) = 1B(z,Ïn)(x)u(tn, x). We then haveu(tn, x) â¥ Ïn(x)210for all x â Rd. Since u(tn, Â·) and Ïn are both bounded (because u â U), it follows from thecomparison principle (recall Remark 4.2.10) thatu(tn + t, x) â¥ uÏn(t, x) (4.7.4)for all (t, x) â Q. Note that Ïn â λδz in MF (Rd), so by Lemma 4.2.6(a) and trans-lation invariance, uÏn(t, x) â uλ(t, x â z). (Recall that uλt (x) = uλδ0(t, x).) Of course,limnââ u(tn + t, x) = u(t, x), and so taking nââ in (4.7.4) we obtainu(t, x) â¥ uλ(t, xâ z).Since this holds for all λ > 0, it follows that u(t, x) â¥ uâ(t, xâ z). The second inequalityin the result then follows from (4.3.5).Proof of Theorem 4.7.3. Let S and µ be as in the statement of the theorem. Suppose thatut(x) is a solution to (4.1.3) in U whose singular set contains S. By Proposition 4.7.4, forsome constant c > 0 we haveut(x) â¥ supzâSctâ1Î² p1(tâ 1α (xâ z))= ctâ 1β p1(tâ 1αd(x,S)), (4.7.5)where the second equality holds because S is closed and p1 is continuous and radiallydecreasing. By assumption, S supports a finite measure µ such that limÎ»ââ uλµt = Ut. Fixλ > 0. By (4.2.16),uλµt (x) â¤ Î»Stµ(x)â¤ λµ(1)pt(d(x,S))= Î»µ(1)tâdÎ± p1(tâ 1α d(x,S)). (4.7.6)Since β < αd , (4.7.5) and (4.7.6) imply that there exists t0(λ) > 0 such thatuλµt (x) â¤ ut(x) for all x â Rd and t â (0, t0(λ)]. (4.7.7)Applying the comparison principle (as in Remark 4.2.10) at time t0(λ), it follows thatuλµt (x) â¤ ut(x) for all (t, x) â Q. (4.7.8)The above holds for all λ > 0. Since limÎ»ââ uλµt = Ut and u â U , it follows that ut(x) =Ut.211It remains to prove our existence result, Theorem 4.1.14(b), which states that if S â Rdis compact and S = supp(µ) for µ â MF (Rd) satisfying (F2)-s for some s < dsat, then thereexists a weak solution of (4.1.3) with initial trace (S, 0) if Î²â(Î±, s) < β < αd . This solutionis uâµt .Proof of Theorem 4.1.14(b). Let µ âMF (Rd) satisfy (F2)-s and have compact support S.By Lemma 4.7.2, uâµt (x) is a weak solution to (4.1.3) and S is contained in the singular setof uâµt . By (4.1.19), uâµt (x) vanishes uniformly on {x : d(x,S) â¥ Ç«} as t â 0 for any Ç« > 0.It follows that for any ξ â Cc(Sc),limtâ0â«Î¾(x)uâµt (x)dx = 0.Hence the singular set of uâµt is no larger than S and (4.1.23) holds with measure ν = 0,which implies that uâµt has initial trace (S, 0).212Chapter 5ConclusionWe conclude with a discussion of our results in a broader context. We also discuss possibleextensions and future work.5.1 Super-Brownian motionIn the chapters on super-Brownian motion, we mentioned that the interest in âZt was dueto an important open problem concerning SPDE: establishing that pathwise uniquenessholds (or does not hold) for the stochastic equationââtXt(x) =â2Xt(x) +âXt(x)WË (t, x), (5.1.1)where WË (t, x) is a space-time white Gaussian noise. We recall that the density of super-Brownian motion in one dimension is the unique in law solution to (5.1.1). It remains unclearhow the local time itself, as constructed in this thesis, could resolve pathwise uniqueness.However, the resolution of the dimension problem is useful here due to the importance ofâZt to this problem and could potentially be applied to proving pathwise uniqueness forSPDE related to (5.1.1).An open question of some interest is whether or not the results about âZt hold forâS(Xt), the boundary of the support. From a geometric standpoint âS(Xt) may be themore natural set. Some natural problems are to determine if âS(Xt) has dimension 2â 2λ0a.s. on {Xt 6= 0}, and if âS(Xt) is a support for Lt. That is, if Lt(S(Xt)c) = 0 almostsurely. Clearly, showing that âZt = âS(Xt) solves both of these problems, since, as werecall from (1.4.7), âS(Xt) â S(Lt) â âZt. This is related to other open problems aboutâZt. For example, it is unknown if âZt can have isolated points. An isolated point in âZtis not in âS(Xt), so the existence of isolated points would imply that the sets are not thesame. Furthermore, since Lt has no atoms, the existence of isolated points would imply213that S(Lt) is strictly smaller than âZt.As we noted in Chapter 1, there has been a parallel investigation into the boundary ofthe range of super-Brownian motion, and in this case the improvement from the boundaryof the zero set to the boundary of the support has been successful [73, 33]. In these works,the zero set and support are defined with respect to the density X(x) of the integratedsuperprocess. (See (1.3.10) and the discussion in Section 1.3.4.) However, the methods inthis case are very different and rely on the special Markov property for exit measures ofsuper-Brownian motion, a tool which does not appear to apply in our setting.It would be interesting to revisit the problems considered in this thesis for super-Brownian motion with (1 + β)-stable branching. We denote this process by Xt and recallthat Xt has a density if and only if β <2d . It is possible to describe the left tail of thedensity in terms of an eigenvalue defined analogously to λ0. However, the dimension of âZthas not been computed, and there has been no attempt to construct the boundary localtime. Despite this, more progress has been made on the problem of pathwise uniquenessin this setting. Yang and Zhou considered the regularity to solutions of SPDE driven bystable noise, including solutions to the SPDEââtXt(x) =â2Xt(x) +Xt(x)11+β FË (t, x), (5.1.2)where FË (t, x) is a one-sided stable noise of index 1 + β. It is known from work of Mytnik[70] that the unique in law solution to (5.1.2) is the density of (1 + β)-stable branchingsuper-Brownian motion. Yang and Zhou [95] proved that pathwise uniqueness is satisfiedfor (5.1.2) when 0 < β <â5â 2. Remarkably, their work did not make use of the so-calledimproved modulus of regularity at the zero set, which has been important in other proofsof pathwise uniqueness (e.g. [74]). It is worth investigating if such improved moduli can beestablished in case of stable noise, and if so, if they could be applied to prove that pathwiseuniqueness holds in (5.1.2) for values of β greater thanâ5â 2.5.2 The (α, β)-superprocessThere are a number of questions related to our work on the (α, β)-superprocess which webelieve merit attention. A very natural problem is to complete the classification of measuresµ for which µ(Xt) > 0 almost surely. In view of Theorems 4.1.9 and 4.1.10, the main problemhere is to determine what happens when S(µ) has dimension dsat = d+Î±â αβ but is Hdsat-null. (Recall that Hδ denotes the xδ-Hausdorff measure, and we showed that µ(Xt) > 0a.s. on {Xt 6= 0} when Hdsat(S(µ)) > 0.) Furthermore, we would like very much to extendthe non-flat result (Theorem 4.1.10) to all measures with compact support of dimensionless than dsat, not simply those whose support is contained in that of an (F2) measure.214One could also consider the case where supp(µ) is non-compact. Our results, as well asinstantaneous propagation, suggest that the local properties, rather than the global ones,determine if µ(Xt) > 0 a.s. or not. However, it would be very interesting to know if thereexist measures for which global properties âoverrideâ local properties, that is if there existsa (potentially infinite) measure µ with unbounded support such that dim(S(µ)) < dsat, butµ(Xt) > 0 a.s. on {Xt 6= 0}. On a related note, Theorem L of [8] implies that for functionsÏ with sufficiently fast growth at infinity, Xt(Ï) = +â a.s. on {Xt 6= 0}. In a sense, thisposits an almost sure global property analogous to the almost sure local properties we haveconsidered.The initial trace theory of the dual equation, that is the fractional PDEââtut(x) = âÎ±ut(x)â u1+β(x), (5.2.1)is still in its initial stages, having been introduced in 2019 [8]. Our results advanced thistheory by characterizing the admissible singular sets in terms of their Hausdorff dimension.Our theorem is not complete, the extensions we mentioned above for our results about Xttranslate to the following in the PDE setting. They are (i) to extend the existence result(Theorem 4.1.14) by removing the condition that S is the support of an (F2) measure toallow S to be a compact set with dim(S) < dsat; and (ii) to understand the âcriticalâ case,that is to determine if solutions exist when dim(S) = dsat with Hdsat(S) = 0.In the super-Brownian case, Le Gall [55] gave a probabilistic characterization of theadmissible initial traces for the dual equation and furthermore gave probabilistic represen-tations for the solutions. This was done using the Brownian snake. Our existence theoremdoes give probabilistic representations; however, while we do show that the admissible sin-gular sets are roughly characterized by their Hausdorff dimension, the admissible initialtraces have not been fully classified, probabilistically or otherwise. An obvious and at-tractive extension of our existence result (Theorem 4.1.14(b)) would be to allow non-zeroregular components. Because the main difficulty in this problem is due to the singularset, we believe it should be possible to generalize the non-singular behaviour without muchdifficulty.It is natural to ask if the snake approach would work in our setting, and if it couldbe used to improve our results. The development of the Le´vy snake (in particular the(1+β)-stable snake) and the associated superprocess representation [18] offers a potentiallypowerful tool for investigating the charging properties of Xt and the initial trace theory ofthe dual equation. A potential obstacle is that Le´vy snakes in general are harder to analysethan the Brownian snake because the height process is non-Markovian. Of course, the issueof âflatnessâ is novel in this setting and does not occur in the super-Brownian case. It is215not clear if the snake approach is well-suited to handling this behaviour.Next we discuss a few problems related to our result on the strict positivity of Xt(x).Recall that we showed that, in dimension one, Xt(x) > 0 for all x â R a.s. on {Xt 6= 0}if β < Î²â(Î±) = αd+α and α > 1 + β, the latter condition implying that Xt is continuous.While this is known to be false for β > Î²â(Î±), the case of β = Î²â(Î±) remains open. Toprove strict positivity in this critical case would require the development of new methods.On the other hand, if it turns out that Xt is not strictly positive, the fractal properties ofits zero set may be of interest.One might also ask what sort of positivity results can be proved when the density is notcontinuous (i.e. when d > 1 and/or α â¤ 1+β). Strict positivity in the sense above cannot beproved here because the density is defined up to Lebesgue null sets. However, the conditionalestimate Lemma 4.4.1 which is central to the proof of strict positivity holds irrespective ofthe regularity of the density, which is why there is hope that an analogous result will holdin the discontinuous regime for β < Î²â(Î±). âStrictâ positivity for a discontinuous densitycould take the following form: for a compact set K â Rd, almost surely on {Xt 6= 0} thereexists Ç« = Ç«(K,Ï) > 0 such that Xt(x) â¥ Ç« for Lebesgue-a.e. x â K.Finally, another direction to consider is the non-absolutely continuous regime. Forβ â¥ αd , the admissible initial measures for (5.2.1) have been exactly classified via Besselcapacities [8]. This is interesting from a probabilistic perspective because it apparentlycharacterizes the measures µ for which it is possible to make sense of the integralâ«dµdXtwhen Xt itself is singular with respect to the Lebesgue measure. (Any work on this wouldrequire defining what is meant by such an integral.) 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A boundary local time for super-Brownian motion and new path properties for superprocess densities
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