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Path properties of superprocesses Tribe, Roger 1989

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PATH PROPERTIES OF SUPERPROCESSES Roger Tribe B. A.. (Mathematics) University of Cambridge A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF MATHEMATICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA June 1989 © Roger Tribe In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of AA/VT̂ gAMrTtCS The University of British Columbia Vancouver, Canada Date DE-6 (2788) Abstract Superprocesses are measure valued diffusions that arise as high density limits of particle systems un- dergoing spatial motion and critical branching.The most closely studied superprocess is super Brownian motion where the underlying spatial motion is Brownian. In chapter 1 we describe the approximating particle systems, the nonstandard model for a superprocess and some known path properties of super Brownian motion. Super Brownian motion is effectively determined by its closed support. In chapter 2 we use the approximating particle systems to derive new path properties for the support process. We find the growth rate of the support for the process started at a point mass. We give a representation for the measure at a fixed time in terms of its support. We show that the support at a fixed time is nearly a totally disconnected set. Finally we calculate the Hausdorff dimension of the range of the process over random time sets. A superprocess can be characterised as the solution to a martingale problem and in chapter 3 we use this characterisation to study the properties of general superprocesses. We investigate when the real valued process given by the measure of a half space under a super symmetric stable process is a semimartingale. We give a description of the behaviour of a general superprocess and its support near extinction. Finally we consider the problem of recovering the spatial motion from a path of the superprocess. ii Table of Contents Abstract ii Acknowledgement iv 1 Construction 1 1.1 Introduction 1 1.2 Watanabe's Theorem 2 1.3 The nonstandard model 9 1.4 Super Brownian motion 12 2 The support process of super Brownian motion 16 2.1 The support process started at a point 16 2.2 Recovering the measure from the support 21 2.3 The connected components of the support 24 2.4 The range of the process over random time sets 32 3 The martingale problem characterisation 46 3.1 The measure of a half space 46 3.2 The death point . 55 3.3 The support near extinction 59 3.4 Recovering the spatial motion . 67 Bibliography 72 iii Acknowledgement I t gives me great pleasure to thank Ed Perkins for acting as my thesis supervisor. He provided a wide variety of interesting problems and constant encouragement throughout my degree. Chapter 1 Construction 1.1 Introduction Consider a population containing a large number of individuals. Each individual moves through space and produces offspring during its lifetime. The rules for the spatial motion and the number of offspring of any individual may depend on its location. To obtain a tractable mathematical model for the behaviour of such a population we make some assumptions. The future motion of each individual depends only on its present location. In particular the motion is independent of the behaviour of the rest of the population. At the end of its lifetime each individual produces a random (possibly zero) number of offspring independently of the other individuals. This model could be used to predict the dispersion of an asexually reproducing species. It also applies to the distribution of a rare gene in a diploid gene pool where the chance of two rare genes meeting and hence interacting can be ignored. Because of the assumption of no interaction between individuals many quantities of interest can be calculated. In Sawyer [21] the distribution and joint distribution of new individuals is calculated under a variety of initial distributions. This thesis studies a continuous limit of such a model in which the mean number of offspring is one. To capture the positions of the whole population we consider the state as a measure consisting of small point masses at the locations of each individual. Then as the population size increases and the- mean lifetime decreases we obtain a limiting measure valued process. The total mass represents the size of the population and the measure of a set A represents the number of individuals situated inside a. In some cases the limit process will have a density which can be thought of as a population density. In passing to the limit some features of the original model are lost. For convenience we shall take the lifetimes of the individuals to be of fixed length but the same limiting process is obtained if the lifetimes are exponentially distributed. The exact distribution of the number of offspring is lost and only the variance is preserved, so we shall take critical binary branching where the number of offspring is 0 or 2 each with probability 1/2. 1 This thesis investigates the limiting measure valued process as a mathematical object and many of the theorems describe properties that are of purely mathematical interest. However the estimates needed to prove these properties might be interpreted to give information about the dispersion of a population. For example in section 2.3 we show that the support of the limiting process where the spatial motion is Brownian is nearly a totally disconnected set. To prove this we break the measure into groups of closely related individuals and estimate the fraction of the poulation that occurs in groups that are isolated from the remaining population. 1.2 Watanabe's Theorem We first give an informal description for a construction of a branching superprocess. We will describe a particle system called a binary branching Feller process which will depend on a parameter /x . The idea is that as we let n increase to infinity the particle system will converge in law to a superprocess. Fix a large integer fi. At time zero we position O(pi) particles in space. On the time interval [0, 1/fi) these particles move independently according to the law of a fixed Feller process. At time 1/fi , for each particle independently we toss a fair coin. If the coin lands tails the particle dies and vanishes. If the coin land heads the particle splits into two. On the interval [l//i,2//i) the particles that are still alive move independently according to the Feller process. At time 2/fi these particles again independently die or split into two. We continue this process for all time. Figure 1.2 shows the evolution for three generations (where we have drawn the particle motion as continuous paths for convenience). One way to keep track of all the particles is to attach mass l / / i to each particle and consider the state at any fixed time as a measure. This measure will be a finite sum of point masses of size l//z. As the branches grow so the measure evolves in time. It is this measure valued process that will approximate a superprocess. We wish to let fx —• co . Notice that the parameter /J has several roles. There are O(fi) initial particles. The mass of each particle is 1/fi so that taking n large and choosing the initial positions carefully we can let the measure at time zero approximate any finite measure. However l//x is also the time between each branching generation. We now investigate the approximating particle systems and in doing so we shall see that a branching rate of 1/fi should lead to a nontrivial limit. The number of particles descended from any one fixed particle is a Galton Watson process. We recall some results from branching processes (see Harris [11] p21-22.) Let (X(n,i) : i,n € N) be I.I.D. random 2 space time Figure 1.1: Binary Branching Feller Process. variables on some probability space (ft, T, P) taking values 0 or 2 each with probability 1/2 . Let Zo = 0 and Zn = £ f = 7 l X(n,i). Then lim nP{Zn > 0) = 2 (1.1) n — • o o Yan^ P(Z„ > nx\Zn > 0) = exp(-2z) for all x > 0. (1.2) We use these results to analyse the measure at a fixed time t > 0. Look back in time a short distance a > 0 . Let I(t,a) be a list of those particles at time t — a that have descendants alive at time t. We consider only those branches between times t — a and t that end with a living particle at time t ( see figure 1.2 ). We see that the mass at time t comes in 'clusters' rooted at points in I(t,a). Each particle at time t — a has an equal and independent chance of having descendants alive at time t. So the number of clusters is a Binomial random variable with parameters (n,p) where n = # of particles at time (t — a) = / ix mass at time (t — a) p = Prob (one particle having descendants time a later) = P(ZafJ > 0) 3 t-a t Figure 1.2: Decomposing the measure into clusters. Now equation (1.1) shows that for large ft, conditional on the mass at time t — a # of clusters « Poisson with mean equal to (2/o)x mass at (t — a) . The masses of particles in each cluster are independent random variables and equation (1.2) shows that for large (i mass of particles in a cluster w Exponential with mean equal to (a/2). Note that if the spatial motion is homogeneous then the exact shape of each cluster about its root is identically distributed. Thus the measure at time t is the superposition of an approximately Poisson number of clusters whose masses are independent and approximately exponentially distributed and which are rooted at points uniformly chosen according to the measure at time t — a. This description becomes more and more accurate as /i —• co and is the basis for many properties of superprocesses. As the parameter fi —* co, if we pick the positions of the initial particles so that the initial measures converge, we hope that the approximating particle systems will converge in law to a finite measure valued process. That this is so and the exact way in which the spatial motion meshes with the branching is the content of Watanabe's Theorem. In the remainder of this section we will develop the notation to state this Theorem.This is essentially taken from Dawson,Iscoe and Perkins [5] section 2. We will descibe a labelling system ( first used by Walsh [24] ) which will allow us to point to any branch on a binary branching Feller process. This labelling system is used extensively throughout this thesis. 4 Notation. E = locally compact separable metric space. £ = Borel c-algebra on E. b£ = bounded Borel measurable functions / : E —• R. Ci = continuous functions / : £ - » R with limits at infinity. Co = continuous functions / : £ - » R vanishing at infinity. M(E) = all measures on (E, S). MF(E) = finite measures on (E,£). M\{E) = probability measures on (E,£). M£(E) = : x , € £ , t f £ N O ) . m( / ) = fE f(x)dm(x) for all m G MF{E),f G b£. f+(x) = f(x)A0. E& = EL) {A} where A is added as a discrete point. N = { 1 , 2 , . . . } No = {0 ,1,.. .} For any metric space M we write D(M) for the space of right continuous paths with left limits mapping [0,oo) —• M with the Skorohod topology and C(M) for the space of continuous paths with the topology of uniform convergence on compacts. We give Mp(E), Mi(E) the topology of weak convergence (which is metrizable). Throughout this thesis C will denote a constant whose exact value is unimportant and may change from line to line. Distinguished constants will be denoted Ci,C2,... There are two underlying sources of randomness for a branching superprocess, a spatial motion and a branching mechanism. As in the above description we shall take the spatial motion to be a Feller process and the branching mechanism to be critical binary branching. This will be sufficient for all the results of this thesis. We shall briefly describe more general branching superprocesses at the end of this section. Let ((YT : t > 0),(PQ : y G E)) be a Feller process with state space E defined on some probability space (QOJ-^O)- Thus (Pg : y G E) is a strong Markov family and its associated semigroup satisfies TT : C0(E) -> C0(E) for all t > 0 \\T,f — /|| —• 0 as tf —• 0 for all / G C0(E). 5 Define A : D(A) C C,(E) — C0(E) by Af(x) = lim(Ttf(x)-f(x))/t D(A) = (/ € C,(E) : lim(Ttf(x) - f(x))/t exists uniformly in x) It follows that D(A) is dense in C\(E). We may extend the Feller process to E& by setting P*(Yt — A , V i > 0 ) = l . Let e be a coin tossing random variable defined on (Sl\,Ti,Pi) taking the values 0 and 2 each with probability one half. Let / = U n eN(N° x {0> }̂")- ̂ he elements of I will label the branches of the branching Feller process. If /?= (/?o, ft ,...,#,•) e / we write \0\ = j for the length of the label 0 . If 0 is of length j then it will label a branch upto time (j + . Write 0 ~ < if < < < (\0\ + 1)/// so that 0 labels a branch upto the first branching time after t. Let 0\i = (0o, • • •,/?«) for i < j .Call /? a descendant of 7 and write 0 y 7 if 7 = for 6ome i < |/?|. Let 0(0,7) = \0\ — inf(j : 0|j ^ 7|j) be the number of generations back that 0 split from 7. Let fi2 = ( D ( £ A ) x {0,1})7, -F 2 = product a-field . Writing u € fi2 as w = ( y a , e a ) a e / we define G„ = <r((Ya,ea) : \a\ < n). Fix n £ N and ( « i ) i 6 N e • W e w i s h t o find a probability P on (fi 2, ̂ 2) which satisfies for any measurable Aa C D(Ent),Ba C {0,1} and all n > 0 P(u» : ( Y a , e ° ) M = 0 E ] j A a x B a ) = JJ ° ( y A ( 1 / M ) € A"). JJ ^(e G B") (1.3) ^ H = 0 |or| = 0 |o| = 0 P{w : (Y°,e°)\al=n € JJ A° x B°|G„)(u;) = TJ P0* "° ( y A ( ( „ + i ) / „ ) € j4 a | y A ( „ / ,0 = Y ^ ^ w ) |cr|=0 \a\=0 x J] Pl(e 6 5 ° ) (1.4) |o|=n By an adaption of the Kolmogorov extension Theorem there exists a unique probability measure P = P!fi)i''> satisfying (1.3) and (1.4). It follows that P(Ya e A) = Po"°(XA(|a|+i)/„ e A) so that each V 0 has the law of the Feller process upto time (\a\ + l)/fi when it is frozen. Also from (1.3),(1.4) (c° : a £ I) are I.I.D. copies of e and are independent of (Ya : a £ I). The ea will indicate whether the particles split or die at the branching generations and this will be independent of the spatial motion. Finally from (1.3),(1.4) the random variables (Yta : |a| = n) are conditionally independent given Gn indicating that the particles move independently between branching times. 6 Let n = E& x n2 = product <7-field. p(*-')..« = siti)i x P 2 ( s > ) i'" Then for u> = ((x̂ ),-, (Y 0 , eQ)ae/) particles will start at those z,- that are not equal to A. Define the death times for the branches as 0 if a 0 = A min((i + : e°'' = 0) if this set is nonempty (|a| + otherwise To each branch a £ / we associate a corresponding particle which moves along the branch until the death time. So the position of the particle on the branch a is given by ( Y" for t < T° Nt°={ [A for t > r a Define a filtration where if j/fi <t<(j+ l)//i A? = a(Ya, e° : |a| < j) V f | a(Yf : |/?| = j,s< «). Also let A1* — \ft>oAt. For Pi,... ,/3n £ I define the information in the branches /?!,...,/?„ by M :«l=l,...,n). Now we attach mass 1/fi to each particle and define a measure valued process TV" : [0,oo) —• Mp(E) by N?(A) = ( l / » x #(/V° 6 A : a ~ t) a~t For f £ b£ v/e write W ) = / f(x)dN?(x) = (l/riJ2f(NtQ) where we shall always take /(A) = 0. Then TV/* e .4, for all t and TV" £ D(MF(E)) almost surely. We shall sometimes need the total mass descended from one branch. Define o~«,a>-/: If trip = (1/A05Z<LI E M£(E) then we extend (xj),<je to (^i) i 6N by setting X* = A for i > K. We write Pm" for p( r>)i.*». This ignores the order of the (x,-)j but note that the order does not affect the measure on ^ ( A ^ : t > 0) in which we are mainly interested. We shall need a strong Markov property . Let T p = (j/n : j = 1,2,...). In Perkins [16] Proposition 2.3. some shift operators are defined and a strong Markov property is proved for stopping times taking values in . (The construction in Perkins [16] for super stable processes is slightly different but the proposition applies here.) Theorem 1.1 (Watanabe [25].) Suppose G M£(E) —» m G MF(E) weakly as \i —* co. Then pM»(N» G •)•-• Qm(-) on D(MF{E)) as ft — oo. (1.5) The law Qm is supported on the subset of continuous paths. Writing Qm again for the restriction to C(Mp(E)) , Xt for the coordinate process and for the canonical completed right continuous filtration then Qm satisfies the following martingale problem Xo = m W) < Xt{f) = m(f) + X,(Af)ds + Zt(f) for all f G D(A) (1.6). Zt(f) is a continuous martingale s.t. (Z(f))t = /J X,(f2)ds The family (Qm : m G MF(E)) is a strong Markov family. We reserve the symbol Qm for the law of the branching super Feller process on path space starting at m. In Watanabe [25] the convergence of the finite dimensional distributions is proved as well as the continuity of the superprocess under some conditions on the semigroup Tt. Also the Laplace functional of the superprocess is identified as follows. For fixed / G D(A) let ut be the unique strong solution of the evolution equation { dut/dt = Aut - u$/2 «o = / Then Em(exp(-Xt(f))) = exp(- J utdm). (1.7) For convergence as distributions on D(M) and for the continuity of the paths of the superprocess in general see Roelly-Coppoletta [19] where it is shown that the law of a Markov process has Laplace functional given by (1.7) if and only if it satisfies the martingale problem (M). From the construction we see that a superprocess inherits from the approximating particle systems the following 'branching' property. If m\,m2 € MF(E) and X},X^ are independent superprocesses started at mi.mj then the process X} + X? has law QM^+M'. One can build branching superprocesses with more general branching mechanisms. If the number of offspring has mean zero and a finite variance <r(x) that depends continuously on the position of the parent x £ E then the convergence in Theorem 1.1 still holds and only the variance a of the offspring distribution enters into the limiting process. Fitzsimmons [10] considers infinite variance branching that depends measurably on position and also spatial motion given by a Borel right process. The correct Laplace functional is formally identified and a measure valued Borel right process that has this Laplace functional is constructed. 1.3 The nonstandard model Why use nonstandard analysis to study a superprocess ? In taking the limit in Watanabe's Theorem we have lost the particle picture. The limiting process takes values in the space of measures. It no longer makes sense to talk of particles dying or having descendants. To do calculations using the intuition of the particle picture we must work with the approximating systems and use weak convergence arguments to obtain results about the superprocess. The idea is to work in the nonstandard universe and to construct a binary branching Feller process exactly as described in section 1.1 but with an infinite branching rate /i."-This will give a process taking values in the nonstandard measures.Theorem 1.2 will show that we can derive from this nonstandard process a standard measure valued process in a very simple manner and that this standard process has the law of a superprocess. Now we can argue using particle calculations on the nonstandard model and transfer results to the superprocess . Many of the limiting arguments seem to be built into the model. Thus nonstandard analysis provides a tool to handle weak convergence arguments efficiently. The nonstandard model was introduced by Perkins and used successfully in Perkins [16],[17],[18] and Dawson, Iscoe and Perkins [5]. We now give an informal description of some definitions and results from nonstandard analysis that we hope motivate Theorem 1.2. Cutland [3] gives an introduction to nonstandard analysis for probabilists which is sufficient for our needs. We start with a superstructure V(S) . S will be large enough to contain the basic spaces for con- structing the binary branching processes i.e. it will contain the reals ,the metric space E, various measure 9 spaces (fioi -̂ o) e.t.c. V(S) is the superstructure obtained by repeated use of the power set operation and is large enough to do any calculations with the binary branching processes . The nonstandard model will live in an extended superstructure V('S). We assume the existence of an embedding * : V(S) —• V('S). Every object in V(S) has an image under * and the embedding satisfies three properties. i. *R is a proper extension of R. This will imply the existence of infinitesimal elements of *R. We will write elements of *R as underscored characters £ , £ , . . . We identify the image of real numbers r 6 R with their images *r . ii. The transfer principle. This allows us to transfer true statements about objects in V(S) to true statements about their images under the embedding. It will imply, for instance, that *R is an ordered field. A £ V('S) is called internal if A £ 'B for some B £ V(S). These are precisely the sets in V('S) that we can describe using the transfer principle. We give one example. Suppose the underlying Feller process was a Poisson process of rate one. Then P0(Yt = 2) = e-'t2/2 for all t £K+ The transfer principle now implies that 'P0('YL = 2) = e-1- f/2 for all t £ 'R+ where we have identified the reals 2,e with the nonstandard reals *2,"e. iii. The saturation principle. This is needed, for instance, in the construction of Loeb measures but its statement would not be helpful here. We can consider the construction of the binary branching Feller processes as a map P : E% x N — Mi(fi) where ((*<),• -> p(x0<.". Under the embedding we obtain a map 'P : ' ( £ ^ x N ) -+ 'M^Q,). So if p £ *N,(xO, £ then is an internal probability on 'A). We also have the embedding of all the particle structure e.g. •N[P £ 'EA for all */? £'I,t£ 'R+ To avoid a notational nightmare we drop the * whenever the context makes clear we are talking about a nonstandard object. For instance we write N[ £ 'EA for all /? £ 'I,t £ 'R+ The transfer principle will allow us to do calculations with the nonstandard branching processes as easily as with their standard equivalents. 10 Call r £ 'R infinite if |r| > n, Vn € N. Otherwise it is called finite. Call r 6 'R infinitesimal if |r| < 1/n, Vn £ N. For every finite r. £ *R there is a unique r £ R such that r — r is infinitesimal. This unique r is called the standard part of r . We write m * r.2 if JJ — r 2 is infinitesimal. Similarly for any metric space M we call mon(j/) = (a; 6 'M : 'd(x, *y) < 1/n, Vn £ N) the monad about y G A/. If x £ mon(y) we call x nearstandard, y the standard part of x and write y = stjv/(x). Let ns(*M) be the set of nearstandard points in 'M. When the space we are working with is clear it is common to write "x for the standard part of x . Indeed for r £ *R we shall write r for the standard part . Let (X, X, v) be an internal measure space i.e. A" is an internal set, X is an internal algebra of sets ( closed under *-finite unions) and v is a finitely additive internal measure on X. ( For example {'E,'£,N^) and (*Q, \4, *Pm") are both internal measure spaces.) Define a real valued set function "v on X by •v{A) = m{v(A)) for all A G X. Loeb showed that the finitely additive measure V has a tr-additive extension denoted by L(u) on the cr-algebra P(X) generated by X. Let L(X) be the completion of <r(X) under L{v). Then (X, L(X), L{v)) is a standard measure space called a Loeb space. If E is a complete separable metric space and u G 'Mp(E) then there are two ways of obtaining a standard finite measure; we may take the Loeb measure L{v) or if u is nearstandard we may take its standard part stMF(E)(v) These are connected by the following result (see Lemma 2 in Anderson and Rashid [2].) v G ns{'MF(E)) if and only if L{y)(ns{*E)e) = 0 and in this case stMF(E){v)(A) = L{u){st-E\A)) for all Ae€. (1.8) Finally we have an elementary nonstandard criterion for convergence in a metric space M . Consider a sequence a(n) £ M as a map a : N —• M . We have an extension *a : *N —• "Af. Then a(n) —• a G M if and only if stM('a(n)) = a for all fi G *N \ N. (1.9) Now we state the main Theorem of this section. Fix n £ *N \ N and let fi = n! ( this ensure that Q C P ) . 11 Notation. We write (*fi,.F, P m ") for the Loeb space ('Vt,L('A), L{'Pm»)) . Also when m„ is fixed we shall often write P, E, 'P, 'E for P m * , Em», 'Pm», *£""" respectively. Theorem 1.2 Let m G Mj?(E) and choose G 'MF(E) so that $tMF(E){mn) = m - Then there is a unique (up to indistinguishability ) continuous Mp valued process Xt on Pm») such that P m " - a.s. X,(A) = L(N[)(srl(A)) for all i 6 n«C[0, oo)), A 6 S. (1.10) Moreover Pm-(X € C) = Qm(C) for all C G B(C(MF(E))). This is nearly immediate from Theorem 1.1 and (1.8),(1.9). For the proof see Dawson,Iscoe and Perkins [5] Theorem 2.3. We shall use the nonstandard model for super Brownian motion in R d throughout chapter 2. In this case there are two very useful results connecting the nonstandard support of the process (Nt_: i > 0) to the support of the process (Xt : t > 0). These are proved in Dawson, Iscoe and Perkins [5] Lemmas 4.8, 4.9. Lemma 1.3 a. For each nearstandard £ G *[0, co) such that t > 0 , with probability one S(Xt) = stR*(S(N£)). (1.11) b. With probability one, for all nearstandard £,£ G *[0, oo) and y ~ t if0<e<t andN? ^ A then °Nl £ S(X,). (1.12) 1.4 Super Brownian motion Super Brownian motion is the most intensively studied superprocess. We give a summary of those path properties that will be used in this thesis. We assume that the process is started at a finite measure. In dimension one the measure Xt has a density X(t,x) which is continuous in (0,co) x R. We shall not consider this case until section 3.1 and delay a careful statement of this result until Theorem 3.1 . In dimension d > 2 the measures Xt are singular with respect to Lebesgue measure for all t > 0. Thus even if the process starts with a smooth density it instantly becomes singular. This result is proved in Dawson and Hochberg [4] for a fixed time and is extended for all times in Perkins [18] in a remarkable way which describes the exact nature of the measure Xt. We explain this result now. 12 For any continuous onto increasing function <j> : [0,oo) —• [0,oo) define a function on the subsets of R d b y oo <f>m(A) = lim inf y^<£(diam(A)) (1.13) d i a m ( D i ) < « , = 1 where the supremum is taken over all countable covers of A C Kd using sets of diameter less than 6. Then <pm(-) is a Borel measure called Hausdorff (̂ -measure. If <$>(x) = xd then <j>m(-) is a multiple of d-dimensional Lebesgue measure. In general however, (f>m() is not a <r-finite measure. If 4>(x)/xd —• oo as x —• 0 it gives a way of distinguishing between d-dimensional Lebesgue null sets. If cj>(x) = xr then <j>m(-) is called Hausdorff r-measure and will give positive measure to smoothly embedded subsets of Rr (e.g. for a smooth curve C , xlm(C) = length(C) ). Define for Borel A € Rd dim(A) = inf(r > 0 : xrm(A) < oo) Then dim(A) , the Hausdorff dimension of A, takes values in [0,d] . Note that for A of dimension r we have i 0 if s > r x'm(A) = { e [0,co] if s = r oo if s < r Notation. For any Borel measure m we write 5(m) for the closed support of m. Theorem 1.4 (Perkins [18]) Let <j>(x) = x2 log + log+(l/a;). Let Xt be super Brownian motion started at m € Mf(R<l) in dimension d > 3. Then there exist constants 0 < Cj < c 2 < oo depending only on d such that with probability one Cl<i>m(A D S(Xt)) < Xt(A) < c24>m(A n S(X,)) Vt > 0, V Borel A (1.14) So, upto a density bounded inside [ci,c2], the measure Xt is a deterministic measure spread over a random closed set S(Xt). This implies immediately that S(Xt) has Hausdorff dimension 2 and hence is Lebesgue null for all t > 0. In dimension 2 there is a less precise result which still implies singularity. Theorem 1.4 allows us to concentrate on the support process (S(Xt) : t > 0) of super Brownian motion. The following two Theorems (proved in Dawson, Iscoe and Perkins [5] Theorems 1.1,4.5 ) show that the support moves with finite speed and gives a modulus of continuity for that speed. Notation. For closed A C Rd and e > 0, let Ac = (x : d(x, A) < s). Theorem 1.5 Let h(t) - ^(log^" 1) V 1). 13 a. For Qm — a.a.u and each c > 2 , 3 6(w,c) such that if 0 < t — s < 6 then b. For each t > 0, for Qm — a.a.u and each c > y/2, 36(u,c) such that if 0 < s < 6 then S(Xt+s) C s(xtyh('\ This Theorem can be derived from a global and local modulus of continuity for the motion of the particles in the nonstandard model. The key is that we can control the motion of all the particles simultaneously. Theorem 1.6 Let h(t) = y/t(\og(t-i) V 1). a. For Pm» —a.a.u and each c > 2, 3 6(u,c) such that i/0 < t — s < 6 for nearstandard s,t £ *[0,oo), P~tandN[j:A, then.\N[ - N[\ < ch(t- s). b. For each nearstandard t G *[0,co] , for P™* — a.a.u and each c > y/2 , 36(u,c) such thai if 0<s<6, P~t + s, N[+L±A then \N£+1-Nl\< ch(t-s). Theorem 1.5 follows from Theorem 1.6 and equations (1.11),(1.12) on the support of the nonstandard model.(The local modulus of continuity is not stated in Dawson, Iscoe and Perkins but the proof is entirely similar and simpler than the global modulus.) From the proofs of these results we also note that Q(%) < P) < <P) = 0 ( p ( ( c 2 / 2 ) _ 2 ) ) as p -> 0 (1.15) Super Brownian motion has a space-time-mass scaling property. For (3 > 0 define Kp : M^(R d ) —• MF(R d ) as follows J f(x)Kpm(dx) — J f((3x)m(dx) for all measurable f. Proposition 1.7 For m 6 Mp(Rd) the law of the process (Xt : t > 0) under Qm equals the law of the process ( /T 1 Kp-i,aXpt • < > 0) under QpK»1"m. For a proof see Roelly-Coppoletta [19] Propositon 1.8. Exact asymptotics for the probability of super Brownian motion giving mass to small balls were proved in Dawson,Iscoe and Perkins [5]. Theorem 1.8 a. For d > 3 there exists a constant C3 G (0,00) depending only on d such that for any 6 > 0 there exists to and \e2-dQm(Xt(B(x, e)) > 0) - c 3 / d pt{x, y)m{dy)\ < 6m(l) + e(c3 + 6) 2m(l) 2/2 14 for all e < e0,t > 6,x G R d , m G MF(Rd). /n particular l rn i ( l / e d - 2 )Q m (X t (B(x , £ ) ) > 0) = c 3 pt(x,y)m(dy) and for any 6 > 0, K < oo the convergence is uniform overt > 6, x G R d ,m(l) < K. b. For d > 3 Mere exists a constant C4 G (0, 00) depending only on d such thai for all x G Rd, £ > The proofs of parts a,b follow from the proof of Dawson,Iscoe and Perkins [5] Theorem 3.1 and part c follows from Evans and Perkins [8] Lemma 1.3 . Finally we state a result on the effect of changing the initial measure on the law of of the process (Evans and Perkins [8] Corollary 2.4). Theorem 1.9 For any m i , m 2 G MF(Rd) and s,t > 0 the laws Qmi(Xt G •) and Qm3(Xs G •) are mutually absolutely continuous. When trying to prove almost sure results about Qm(Xt G •) this will allow us to choose m and t > 0 at our convenience. 0, t > e2, m G MF(Rd) c. There exists a constant C5 G (0,oo) depending on d such that for all £,t,m,x 15 C h a p t e r 2 The support process of super B r o w n i a n motion 2.1 T h e support process started at a point For super Brownian motion started at a point mass the growth of the support is controlled by the local modulus of continuity Theorem 1.5b. We show that in this case there is a limit result for the rate o f growth. Theorem 2.1 Lei g(t) = y/2t log^"1) and p(t) - inf{r : S(Xt) C 5(0, r)}. Then For Q6a - a.a.u : lim 44 = 1 First we reinterpret the classical results on Galton Watson processes in (1.1),(1.2) in terms of the nonstandard model. Lemma 2.2 For nearstandard £, £ > 0 such that x,t > 0 and y ~ t a. /i -P"" l *°(jV£(l) >0) « 2*-1 b. P»~ls°(N£(l) > *|JV£(1) > 0) = e-( 2*/') c. P"~ls°(Nl{l) > x\Nl ± A) > 2XI" 1 exp(-2z/t) PROOF OF LEMMA 2.2. Parts a,b follow from (1.1),(1.2) and the transfer principle. For part c, f i x j G {0,1,... ,2"'}. Then a counting argument shows -P"~lto(liN£(l) = j, Nl^A) = j2~>'t'Pti~ls°(LiNl = j). Sofor £G{0 )l//i,2//x>...,2'"/^} •P"~1So(N^(l) > x_\NZ ? A) = 2"<'P"" ,*0(/V7'(1) > x, Nf ^ A) 2"* = £ rp>i~l6o((*N£(i) = j) > nx'P^^iN^l) >£) = xln'P^^iNlil) > 0)yP"~le°(Nl(l) > x\N£(l) > 0) 16 and the result follows from parts a,b.D PROOF OF THEOREM 2.1. The local modulus of continuity (Theorem 1.5b ) implies it is enough to show For Qs° - a.a.u : lim inf > 1 t-o g(t) ~ We use the nonstandard model for super Brownian motion taking x< = 0 , i = 1,..., fx so that the initial mass equals 6Q. Fix 9 G Q , e G R such that e,0 € (0,1).Define A , = {W : s u p ( | < | : 0 ~ 6n) < (1 - CM*")} Recall that I(9n,9n) lists the particles at time zero that have descendants alive at time 6n. For each 7 G 1(9",0") pick 7 ~ 6n such that 7 -< 7 and ^ A. P{An) < p{sup(\N^\:7eI(9n,9n))<(l-£)g(9n)) = E TT -P(\Nl\<(l-e)g(6n) _7€/(»",»") = £?[( lW n )*<'" ' ">] where J„ = Po(p3s»| > (1 - e)g(9n)) and Z(0n,9n) is the cardinality of I(9n,0n). Lemma 2.2 shows that Z(9n,9n) has a ""-binomial distribution B(n,p) with n = fi and /ip w 20". Using the bound Po(\Bi\ >x)> C(d)xd-2e-x3'2 for x > 1 we have P ( A n ) < ° ( ( l - / n p ) " ) = exp(-2<r n/ n) < exp (-Cnd-20-cn) Borel Cantelli implies there exists N(u>) < 00 almost surely such that For all n > N(u), 30 ~ 9n such that |JVf. | > (1 - e)ff(0") Now fix w such that N(u) < 00 and outside a null set so that the support relations (1.11),(1.12) and the modulus of continuity for particles Theorem 1.6a holds. Choose n > N(u>) such that 0n < 6(u>,3). Find 0 ~ 6n such that |JV^„| > (1 — e)g(0n) .The modulus of continuity implies > (l-e)g(0n) - 3^"(1 - 9W\og((9»(l - for 0n+1 < i < 9 n 17 > g{6n) ( l - z - (3/V2) ((1 - 0)(1 + log(l - <?)/nlog(0))1/2) > g(t) (l - e - (3/v/2) ((1 - 6)(1 + log(l - <?)/nlog(0))1/2) But -jVf G S(Xt) by (1.11) so liminf 44 > 1 - e - 3>/l - 0 wo <,(<) - Now take sequences e„ | 0, 9n T 1 to show that with probability one l iminf44>l Finally we note that the set {u : liminf(_o/>( l)/fl'(0 ^ 1} * s Borel in C([0, oo), M f ) and so we may transfer the result to path space.• • Thus for small t the support is approximately contained in a ball of radius g(t). We may normalise the support so that it has radius one. We show in Corollary 2.5 that as t —• 0 this normalised support 'fills' the whole unit ball. The following Lemmas examine how fast holes appear in the support. Lemma 2.3 For d > 3 , 0 < r < l a. If k < (1 — r2)/(d — 2) then with probability one lim sup d(x,S(Xt)/g(t)) = 0.- <-*0+ x£B(0,r) b. If k > (1 — r2)/(d — 2) then with probability one limsupt -* sup d(x, S(Xt)/g(t)) = oo. t—0+ * € B ( 0 , r ) P R O O F , a. Fix r € (0,1), ib < (1 - r7)/(d- 2). We look for holes inside S(Xt) n B(0,rg(t)) of size <*+(i/2). D e f i n e Grid= ((t )z):r = n - 1 / 3 , n = l , . . . , z £ (tk+W/2y/d)zd,\x\ <rg(tj) We first show that for small t there are no holes centered at a point of the Grid. For (n - l y , 3 ,x ) G Grid we have Q ^ ( x n - l / 3 ( 5 ( x , ( l / 4 ) n - ( 2 « : + 1 ) / 6 ) ) = 0) = g " ' / 3 < ° (x 1(5(n 1/ 6x,(l/4)n-*/ 3)) = 0) 1/3 = [ l -Q < o (x 1 (5 (n 1 / 6 x , ( l / 4 )n - f c / 3 ) )>0) ]" 18 ,1/3 by first the space-time-mass scaling and then the branching property of super Brownian motion. From Theorem 1.8c Q6° (xl{B{nllex,{\/A)n-kl3)) >u) > C{n-k^l3Vl{nl^x) A 1). Pick e such that k < (1 — (1 — e) _ 1 r 2 ) / (d — 2) and choose «o such that (l/4)n~k*3 < e. Noting that l6x\ < ry/{2/i) log n we have for n> n0 Q6° (xn-i/3(5( a:,(l/4)n-( 2 , : + 1)/ 6)) = 0) < [ l -Cn-*( d - 2 ) / 3 exp(-r 2 log(n) /3( l -e) ) ' < expC -Cn^/ 3 ) -^- 2 ) / 3 -^^ 1 - ' ) ) Therefore J2 Q(°(Xt(B(x,(l/4)ik+W)) = 0)<J2 Cnkd'3(logn)dexp(-Cn^-k^-r3^1-^3) ( t , x ) € G r i d : ( < n - 1 / 3 This sums over n and Borel Cantelli implies 3N(u>) < co almost surely such that if (t, x) G Grid and t < N~1/3 then Xt{B(x, (l/4)tk+W)) > 0 (2.16) Fix u such that N(u>) < co and off a null set so that the global modulus of continuity (Theorem 1.5a) holds.Now argue by contradiction. Suppose 3< < min(6(w,3), N'1/3,2~20) and an x G B(0,rg(t)) such that X,(B(x,tk+W)) = 0 .Pick n> N such that (n+1)" 1/ 3 < t < n" 1 / 3 . We use the global modulus of continuity for the support to show there must be a hole (of smaller radius) centered at a grid point.The modulus of continuity implies * „ - , / , (B(x, t k + W _ 3/ l((n)"1/3 - (n + l)" 1' 3))) = 0 But * * + ( l / 2 ) _ 3 / l ( ( n ) - l / 3 _ ( n + l ) " l / 3 ) > „-(a*+l) /6 _ ( 3 n - 4 / 3 l o g ( 3 n 4 / S ) ) l / 2 > (l /2)n-( 2 f c + 1 )/ 6 so that X „ - I / 3 ( B ( x , ( l / 2 ) r z - ( 2 t + 1 ) / 6 ) ) = 0 But we can find x0 such that (n~^3,xo) G Grid and B(x0, (l/4)n-( 2* + 1V 6) C B(x, ( l /2)n-( 2 l + 1 ) / 6 ) . Thus X B. I /,(fl(x 0 ) (l /4)n-( a *+ 1 )/ 6 )) = 0 19 which contradicts equation 2.16 . So for small t , for all x G 5(0, rg(t)) we have t~k-^l2^d{x, S(Xt)) < 1. Thus limsupi- t(21og(i- 1)) 1/ 2 sup d(x,S(Xt)/g(t)) < 1. *->0+ x€B(0,r) But k < (1 — r2)/(d — 2) was arbitrary and the result follows. b. Let xt have coordinates (rg(t),Q,... ,0) for f < 1. Then using Theorem 1.8b Qto(Xt(B(xut'))>0) < Ctl^p(t + t2t,rg(t)) < C - < / ( d - 2 ) - ( d / 2 ) + ^ : ' ( l + ( 3 , - 1 ) - 1 If / > (1/2) + (1 — r2)/(d — 2) this probability tends to zero as t -+ 0 so that along a fast enough sequence t„ — 0 Borel Cantelli guarantees Xtn(B(xtn,(tn)')) = 0 for large n. So if k > (1 - r2)/(d - 2) limsup i- l(21og(<-1))1/2 sup d(x,S(Xt)/g(t))>l. t-»0+ *6B(0,r) But > (1 — r2)/(d — 2) was arbitrary and the result follows. • In dimension 1 and 2 we do not have estimates on the probability of charging small balls given by Theorem 1.8.The following Lemma gives such an estimate and hence an upper bound for the rate at which holes appear in the support in these dimensions .While this bound is certainly not best possible it will be sufficient to prove Corollary 2.5. Lemma 2.4 a. For all x G Rd,t > 0 Qs°(Xt(B(x,e)) = 0) < exp (-2(2x) d / 2 £ d I -( d + 2 ) / 2 exp(- l - 1 ( | |x | | + £) 2)) b. For = 1,2 and 0 < r < l if k < (1 - r2)/d then with probability one lim t~k sup d(x,S(Xt)/g(t)) = 0. t _ ' 0 + *€B(0,r) PROOF. a. It will be enough to prove this for the nonstandard model with x, = 0, i = 1,... I(t,t) lists the particles alive at time zero that have descendents alive at time t. For each y G I(t,t) pick 7 ~ t such that y -< 7 and N? ^ A. Using (1.11) we have P(Xt(B(x,e)) = 0) < ? f | (N?tB(x,e)) \76/(*,0 / = E [(1 - /,)*<*•'>' 20 where It = Po(Bt 6 B(x,e)). Z(t,t) has a *-binomial distribution B(fi,p) where up s» 2t 1 . So p{xt(B(^r)) = o) < = exp(-2<-17t) < exp (-2(27r) d/ 2e dr< d + 2>' 2 exp(-r HlMI + E ) 2 ) ) The bound is continuous in e so we may replace B(x,e) by B(x,e). b. We follow the proof of Theorem 2.3a. replacing the equation 2.16 using the bound above by Q6° (* n - 1 / 3 (f l (x,( l /4)n-( 2 i + 1 ) / 6 )) = o) < e x p ( - C n - « 2 * + 1 ) d - ( d + 2 ) + 2 r 3 ) / 6 ) The remainder of the proof carries through.• The Hausdorff metric on compact subsets of R d is defined as follows. For K\,Ki nonempty compact sets dH(Ki,K2) — max( sup d(x,K2) A 1, sup d(x,Ki) A 1) dn{Ki,9) = 1 Combining Theorem 2.1 and Lemmas 2.3,2.4 we obtain Corollary 2.5 If d > 1 then with probability one dH(S(Xt)/g(t), 5(0TT)) -> 0 as t - 0 + . 2.2 Recovering the measure from the support For a fixed time t > 0, Theorem 1.4 can be improved as follows. For d > 3 there is a constant c6 depending only on the dimension so that with probability one Xt(A) = c6<t>m(A n S(Xt)) for all Borel A. The proof, due to Perkins (private communication), uses the 0 — 1 law explained in Proposition 2.11. Thus for d > 3 and fixed t > 0 the measure Xt can be completely recovered from its support. We now give an alternate method for recovering Xt from its support. It is an analogue of a Theorem of Kingman on Brownian local time.Let l(t, x) be the local time of a Brownian motion Bt. Let Z(t, x) — {s < t : B, = x) . Recall that for a closed set A we write Ae for the set {x : d(x, A) < e}. We also write 21 Leb(A) for the Lebesgue measure of a set A. In Kingman [13] it is shown that there exists a constant C7 such that for fixed x,t with probability one lim <T1/2 Leb( Z(t, xY) = c7l{t,x). Theorem 2.6 For d > 3,< > 0 and Borel A of finite Lebesgue measure, with probability one e2-dLeb(S(Xty n A) £ c3Xt(A) as e — 0 where c3 is the universal constant occuring in Theorem 1.8 a. PROOF. Fix t> 0 and Borel A. Let K\ = e2-dLeb(S(Xt) D A). We shall show limsup£((J<0 2) < c\E{{Xt{A))2) (2.17) lim E(KctY) = c3E(Xt(A)Y) for all Y G I 2 ( V (2.18) The result then follows for lim sup E((Ket - c3Xt(A))2) = l i m s u p ( £ ; ( ( ^ ) 2 ) - 2c3E{K[Xi{A)) + c2E((Xt(A))2)) < 0 ' since t —* Xt(A) is continuous and hence Xt(A) G L2(\fs>0JRT-6)- Proof of (2.17). We shall prove this for the nonstandard model. E{{K't)2) = e4~2d J P(Xt(B(x,e))>0,Xt(B(y,e))>0)dxdy. (2.19) Ax A To calculate the probability that occurs as the integrand in (2.19) we use the following idea. Recall that the support moves with a modulus of continuity given by ch(t) where c > 2. We shall choose a suitable value of c later. Using the notation of Theorem 1.6 let Ga = (u> : 6(u,c) > a) be the set where the global modulus of continuity for particles holds for time intervals less than a . For a path in G« , the only particles that can enter B(x,c) at time t must lie in B(x, e+ ch(6)) at time t — 6. So if the distance between B(x,e) and B(y,e) is at least 2ch(6) then on the set d , if we condition on the measure Xt-s, the events (Xt(B(x,c)) > 0) and (Xt(B(y,e)) > 0) are independent. 22 Fix a > e2,6 6 ( £ 2 , a ) and x,y such that \x — y\ > 3c/i(i5).We write m\g for the measure m restricted to B. Using Theorem 1.8a we can find Co such that for all £ < £o e4-2dP(Xt(B(x,e)) > 0,Xt{B(y,e)) > 0,Ga\o-(At-6)) < e 4 - 2 d P x ' - s l e ( ' . ' + ' M < ) ) ( X i ( 5 ( x 1 £ ) ) > 0)Px<-^B(»-'+°w>(X6(B(y,£)) > 0) < e4-2dPx*-'(X6(B(x, e)) > 0)Px^(Xi(B(y,e)) > 0) < ^c3Jps(x,z)Xt^(dz) + 6Xt-l(l) + £(c3 + 6)2Xl((l)/2^ x (c 3 Jps(y,z)Xt.((dz)-r6Xt-6(l) + e(c3 + 6)2X2_s(l)/2^ < c2 Jp6(x,z)Xt_s(dz) Jps(y,z)Xt_6(dz) + C(6 + e)(l + X4_6(l)) So for e < £o e4~2d J J P(Xt(B(x,e)) > 0,Xt(B(y, £)) > 0, Ga)dxdy AxAn(\x-y\>3ch(S)) < c2E(X2(A)) + C(6 + e)\A\2(l + E(X4_6(l))) (2.20) Similarly using Theorem 1.8b and conditioning on o-(^r(_ta) e4~2d J JP(Xt(B(x,e)) >0,Xt(B(y,e)) > 0,Ga)dxdy A x A n (\x - y\ < Zch(6)) < e4~2d J JP(Xt(B(x,e)) > 0,Ga)dxdy 'Ax Ar\(\x-y\<3ch(c2)) +£4~2dE( J JPx<-'*(Xc>(B(x,e))>0)Px<->(Xc3(B(y,e))>0)dxdy) A x A n (|x - y\ < 3c/i(<5)) < Cm(l )m£ 2 - d (£ log( l /£ ) ) d + C|yl|(/ l(6)) di;(X 2_ t 3(l)). (2.21) Combining (2.20) and (2.21) we see that limsup£((/<7) 2iGj < c\E{X2(A)) + CS t—o where C depends only on m, d,t,A. However 6 < a was arbitrary and so limsup E((K')2la,) < c3E(X2(A)). To remove the restriction to Ga we note that P(Ga) —> 1 as o | 0 and so it is certainly enough to show sup £ > 0 E((K')4) < oo. £8-4d J J J Jp([[Xt(B(xi,e))>0)dx1...dx4) AxAxAxA 1 = 1 (\xi - X j \ > 3c/i(£ 2) : i^j) 23 AxAxAxA ^ , = 1 ' {\Xi-Xj\>$ch{t*):ii:j) < C\E{{J Pc,{aiz)Xt_0{dz))^+Ct6-id^-A using equation (1.15) . Fixing c so that c2 — 4 > Ad — 8 this expression is uniformly bounded in e. The other regions of AxAxAxA give smaller contributions as in the derivation of (2.21). Proof of (2.18). Fix 6 > 0 and C G ?t-6- Using Theorem 1.8a l im£(I c *7|^-<) = \im£2-dlc [ Qx'-6(X((B(x,£))>0)dx c—0 e - 0 J A = c 3 I c / / V6{x,y)Xt-t{dy)dx JA JR" By the uniform integrability of the K\ \imE(lcK<t) = E(c3lc f v6(y,A)Xt.6(dy)) = c3E(lcXt(A)). Since sup e > 0 E((K')4) < oo we may extend to all Y G L2(\J6>0 Tt-s). O 2.3 The connected components of the support The arguments that lead to the upper bound on the Hausdorff measure of the support use covers of the support that have a Cantor set like structure. Don Dawson asked the following question: For fixed t > 0 , is S(Xt) a totally disconnected set ? We now prove the following partial answer. Theorem 2.7 Let Comp(x) denote the connected component of S(Xt) containing x. If d > 3 then for all m G MF(RD) and t > 0, with probability one Comp(x) = {x} for Xt — a.a.x. Notation. For t. a. 6 G T" . 8 ~ t let 7~l+ 2 . 3 . 7 > - / J 24 Z^(a) is the mass of the 'cluster' of particles descended from that are alive at time t + a 2 . The following lemma shows that there is a good chance ( independent of a ) that these particles have not spread more than a distance 0(a) from their common root. For m 6 Mp(Rd) choose 6 'Mp(Kd) such that s < M F ( m / i ) m - Lemma 2.8 For nearstandard a, 0_€ *[0,co) such that a = "a > 0,0 = °0_ > 0 we Aaue Pm"(W^(a,i) = 0\Zp(a)>0) = p(0) (2.22) £m" {Z0(a)\(W0{a,0) = 0)|Z"(a) > 0) = r(0)a2 (2.23) where if 0 € R, 0 > 0 Men p(0) > 0, r(0) > 0 and QW°(Xl(B(O,0y) = 0) = exp(p(0)-l) (2.24) £?d/»)«-(X 1(fl(0 )fl))I(X 1(S(0,fl) e) = 0)) = r(0)exp(p(0)-l) (2.25) PROOF. This is essentially due to scaling (Proposition 1.7.) Fix nearstandard a ,£ such that a,6 > 0. For 0 such that /?|0 7^ A define p(a,S.) = 'Pm"(Wp(a,0) = O\Z^(a)>O) r(a,S) = 'Em*(Zp(a)I(W0(a,l) = O)\Zp(a)>O) The values of p(a,0), r(a,0) do not depend on the choice of or /?.Take x, = 0 for i = 1,..., [/ia2/2] and Xj = A otherwise , so that s< m p (m / J ) = (a2/2)6o- Then Q ( I / 2 ) « O ( X I ( S ( 0 QY) = o) = g ( ° a / 2 ^ o ( X a 3 ( f l ( 0 , a ^ ) c = 0) = Pm»{Na'(st-1(B(O,a0)c)) = O) = Pm'(Ne?(B(0,aB)e) = 0) using scaling and the fact that Xi(dB(0,r)) = 0, almost surely for any r. So Q( 1/2)«o( J f l(B(0,fl) e) = 0) [ ^ 3 / 2 ] = ^m"( n (wr'(fi>£)=°)) 1=1 25 . JJ -Pm"(Wx-(a,0) = O) 1=1 > JJ ('Pm«(Wx'(a.,£) = 0\ZXi(a) > Q)'Pm>{ZXi(a) > 0) + 'Pm"(ZXi(a.) = 0)) •=i (l + (p(a,l)-iyPm-(Zx\a)>0)) = e x p ( ° p ( a , £ ) - l ) t ^ 3 / 2 ] l since [/ia2/2]*Pm"(Z j ; ,(a) > 0) ~ 1 from Lemma 2.2a. So °p(a,£) is constant in a and (2.22) and (2.24) follow taking p(6) = °p(a,6). Similarly E6°'2 (x1(B(o, emx^Bio, ey) = o)) = EA'6°>2 (a- 2 X a , (5(0, af?))I(X a »(B(0,a0) c ) = 0)) / Irt'M t^a/2] = (a" 2 )' l'E( £ ZXi(*W fl ^ i ( ' l ^ ) = 0)) \ i = l ; = 1 / W / 2 ] [/.a3/2] = (a_2)° E (̂̂ *(a)l̂ i(a,£) = 0)-P( f| Wx>(a,e) = 0) •=i [^3/2) a- 2exp(p('?))-l)r[ W(a)I ( W r*'(fl . f i ) = 0)|Z*'(fl)>0) •=i x*P(Z I i(a) > 0 ) / * P ( W r ' ( a , £ ) = 0)] a-2exp(p('?)) - 1))° (>a2/2]-P(Z"(a) > 0)r(a,£)/(l - (1 - p ( a , £ ) ) ' P ( ^ ( a ) > 0))) a-2exp(p(0))-l)rr(a,£) So (a - 2 ) °r(a,£) is independent of a and (2.23) , (2.25) follow taking r(6) = (a - 2 ) °r(a,£) . Finally Q{°/2(A:i(Rd) = 0) = exp(-l) so that p(6), r(6) > 0 will follow if we can show QSo'3(Xi(B(0,0)e) = 0,X1(Rd) # 0) > 0 But since the support of the process moves with finite speed , for small enough s we have QSo/2{x,(B{o,ey) = o,x,(Rd) # o) > o and Theorem 1.9 implies (2.26) holds.• Notation. (2.26) 26 For a G (0, co) define Qa = {(y,r,S) € Qd x Q x Q : r,6 > 0 , \y\ < r - 6 , \y\ + r + 6 < a} For a G [0,co), m G Aff(R d ) define Ann(m,a) C R d by Ann(m,a) = (̂ J | x 6 R d : m(z : r — 6 < \z — x — y\ < r + 6) | (y,M)eQa For a € *[0,co), m G *M F (R d ) define Ann(m,a) C 'Rd by Ann(m, a) = | x G *R d : m(z : r — <5 < |z — x — y\ < r + <5)| (y,r,«)eQ« Ann(m,a) is denned so that for x G Ann(m,a) there is a mass free annulus of positive standard rational thickness that disconnects x from J9(x, a) c. Note that if x G Ann(X t ,a) then Comp(x) C 5(x,a) . Thus oo x G j^l Ann(X<,n - 1) Comp(x) C {x} n = l For t,a,9 G T", /? ~ t ,N[ ^ A define ^ ( a , £ ) = p" 1 £ I ( | / V 7 + f t a _ iVf | < 2a£) 7~l+£2,7>i0 Note that if a,9_,t G T' 4 , a > 0, 0 < 9 < 1/2, /? ~ £ are such that A^f_o3 £ A , W l ± - ^ ( a , 0 ) = V V ' - - i a ( a i £ ) = 0 then there is a particle free annulus surrounding Nf N[(z: 5 a £ / 4 < \z - N^_^\ < 7a 9/4) = 0 \N[-Nl_^\<a0 We may shift the annulus slightly to be centered at a rational and have positive rational thickness s o that N[ G Ann(Ar/\a). The following lemma shows that on average a positive fraction of the initial particles will lie inside Ann(Wt",&). Lemma 2.9 For nearstandard a,t G with 0 < a < 1, 2a 1 / 3 < t < co, d > 3 if stM(m^) — m G Mp(R d ) Men Mere exists a constant p > 0 depending only on d such that Emu j ^ T i ^ G Ann ( / V £ , f l ) , A £ ± A) j > />m(Rd) 27 PROOF. The remark following the definition of V 7 ( a , £ ) shows that E ( ^ 32 W e Ann(/V^, £), TV7 # A ) 7~t > £ I//"1 32 1( NL * A - w n ' 1 1 " - = 0, V 7 l i " ^ ( a , £ ) = 0) 7~l = E ]T ^ ( a ) i ( ^ ( a > £ ) = ^ ( a , £ ) = 0) \7~i-a:' / = £ [ ] T • £ ( ^ ( . a ) I ( ^ ( a , £ ) = 0)|A_ a O*-P(^(a,£) = OM 1_ aO \7~i-a3 (2.27) since conditional on At-^ the random variables V(g.,6_) and ^ 7 ( f i t ) I (W 7 ( f l ,£) = 0) are ""-independent. Now using the *-Markov property (see Perkins [16] Proposition 2.3. ) "P (vn,(fl,fi) = o)M 1 _ 4 0 _ . .p 1-M. i - * 3 ( jV>(B(- ,2a£)) = 0) = Q*.-.3(Xa 3(B(-,2a<?)) = 0) | 7 •< vt-o» where in the second step we used the continuity in 0 of Qm(X,(B(x,2a9)) = 0) and the fact that stM(N^_3 - n~16Ny ) = Xt-a2. Also -75 ( ^ ( a ) I ( W " ( a , £ ) = 0)H t _ f t 0 = "2? ( / ^ ( a ) I ( ^ ( a , £ ) = 0 ) | A £ f t a # A ) I ( A £ £ , ^ A ) = • [ • £ ( ^ ( f t ) I ( W n , ( f l , f i ) = 0 | Z 7 ( A ) > 0 ) / i - P (z^Ol) > 0 | A ^ _ A A # A ) ] I(iV7_a, * A ) = 2r(0)I(Ar 7_ £, # A ) using Lemma 2.8 and Lemma 2.2. Substituting into equation 2.27 we get E\^32 W € A n n ^ . o ) , TV7 ^ A ) ] 7~1 > 2r(0)£ Z^"1 £ * A ) Q ^ . - 3 2a* ) ) = 0) 7~i "AT 7 = 2r{0)E Qx'-3(Xa,(B(x,2a0)) = 0)dXt_a,(x) 28 > 2r{9)E ( l - c4(2a9)d-2(2ira2(l + 492))~d'2 . J exp(-(* - y)2(2a2(l + 402))-1)dX(_o,(y)) dXt_a,(x) using the estimate on hitting balls in Theorem 1.8b. Lemma 2.10 gives an estimate on the expectation of this double integral and leads directly to E ^ £ I(A£ € Ann(/V/\a), JV? # A) j > 2 r ( e ) m ( R d ) ( l - C T d - 2 ( l + 4e 2 ) 1 - d / 2 (2 d / 2 - r m(R d ))) Now take 9 > 0 small enough so that the right hand side is strictly positive. • Lemma 2.10 IfQ < a < l,t > al'3,d> 3,m € M F ( R d ) then Em | y Jexp(-(* - y)2/2a)dXt dXt < am(R d)(2 d' 2 + m(Rd)) PROOF . Let pt(x) be the Brownian transition density with associated semigroup Pt. We have for positive measurable f,g (see Dynkin [6] Theorem 1.1) Em [Xt(f)Xt(g)] = JPtf{x)dm(x) JPtg{x)dm{x) + j dm{x) J* Pt_,{P.f P,g){x)ds By approximating positive measurable h(x,y) by functions of the form YA=I f>(x)9i(x) w e have Em[J J h(x,y)dXt(x)dXt(y)] = Jdm(Zl) Jdm(z2) Jdx Jdy h(x,y)pt(x - Zl)pt(y - z2) + Jdm{x) J^ds Jdyp,{x-y) Jdzx jdz2p,{y - Zi)p,(y - z2)M2i> z2) < (m(R d)) 2 sup E \h{B], B2)] + m(R d) f sup E [h(Bl, B2)] (2.28) 2 l , 2 3 JO 2 i , 2 2 where B},B2 are independent Brownian motions satrting at zi,z2.Foi h(x,y) = exp(—(x — y)2/2a) we have sup E \h{B],B2,)) = /exp(-i 2/2a)p 2,(x)da; < {2a/a + 2s)d/2. Substituting into equation (2.28) and using the bounds on a and t gives the result. • Proof of Theorem 2.7 We prove the result first for the nonstandard model with x^ = 0 for i = 1,. . . / i , xt = A for i > \x so that if = fi'1 £ , -6*,I(z i ? A ) t h e n Pm"{X G •) = QS°(X G •)• F r o m 29 Lemma 2.9 we have for nearstandard 0 , ( 6 ^ such that 0 < a < 1, 2a 1 / 3 < t < co 0<p < E I (i-1 W G Ann(N[, a), N? ± A) = E £* P(*L e Ann(A^,<i)|A£ ^ A)I(A? ^ A) = P {Ni G Ann(A^,a)|A£ ^ A) where P(-\N^ ^ A) is the Loeb measure induced by *P(-\N"£ ^ A) . Since Ann(N£,a) decreases as a decreases we have for any 7 ~ t 00 P(Nl G f | Ann(A^, n " 1 ) ^ ^ A) > p > 0 (2.29) n=l We now use a zero-one law to show this probability is in fact 1. Notation. Fix 7 ~ £ . For a,u G T' 1 , < £ define W** = V(JV> - A r t V M ^ 7 ) : a i < / ' - V ( / ? I 7 ) < £ ) V V ( A r t -A r7- i =Ii<l<i i) w t = y w n - i j , n Wo+ = f | W „ - i n The following two results are due to Ed Perkins ( personal communication.) P r o p o s i t i o n 2.11 For A G H0+, P{A\N£ ^ A) = 0 or 1. P r o p o s i t i o n 2.12 IfO<a< 2~ 4 / d then P(-|A£ ^ A) almost surely , for r small enough d ({N[ : ̂ -V(/?,T) > (2r)«}, A?) > r For {y,r,6) G Qa, u,v£ , u < £ define r y , r A«,«i(7) = {u:\Nl-NZ-y\$(r-6,r + 6)foTal\psl.u<Li-1o-(p,1)<v} OO OO OO r W = U f l U f l r v , rA i - , t - ' (T)e7(o + i = l n = l (s/,r,«)€Qn-i i = t + l If w G HnLii^i Ann(A 7 i ,n _ 1 ) then w G T(7) and so equation 2.29 and the zero-one law imply P(n7)\N?_ ± A) = 1. Let A(7) = {w : for small r , d{{N[ : LI~1CT(P,7) > (2r)°} , A£) > r} 30 so that Proposition 2.12 says P(A(y)\N? / A) = 1. Then E ( n" 1 £ I(u» g r(7) n A(7), A? # A) = ^ (v'1 E * F ( W * r(T) n A(7)|/V^ # A)I(A£ # A) = p ( w ^ r( 7)nA( 7)|/v i 7^A) = 0 From the global modulus of continuity for particles, with probability one all the particles move only an infinitessimal distance in an infinitessimal time. So equation (1.11) and the above imply we can pick a single P null set such that if u> £ N we have simultaneously For all nearstandard £ < £ and /? ~ 1, Aff ̂  A we have iVf w A f (2.31) st(5(ATi)) = S(X«) (2.32) Now fix ui g N, 7 ~ t such that N£ ̂  A, w 6 T(7) p| A(y) . We claim oo ° NL G fl A n n ( ^ , n _ 1 ) n = l To show this find k so that oo oo w e n u n r ^ A ; - 1 . * - ^ ) n = 1 (»,<-,«)e<3„-i j=*+i Find r 0 such that (2r 0 ) 1 / 2 < fc-1 and d({N[ : fi~la(P,y) > (2r 0) 1/2}, A?) > r 0 (2.33) Pick n so that n - 1 < ro and find (y, r, 5) £ Q „ - i so that oo " 6 fl r V i r > 4 j- l i t-,( 7) (2.34) 31 For 0 ~ t such that Li- 1o-(P,y) > (2r 0 ) 1 / 2 equation (2.33) ensures that \N[ - AT7| > n'1 and so Wl-iq-y\^(r-6tr + S). For P ~ t such that 0 < °(/i - 1(r(/?, 7)) < fc-1 equation (2.34) and the definition of Tyri j-i ^ - 1 ( 7 ) ensure |Arf - A^7 - y\ & (r - 6, r + 6). For P ~ < such that fi~1a(P,y) w 0 equation (2.31) ensures Aff ss N 7. So { 2 G •R d : r - 6 < \z - A? - y\ < r + n {jV* : 0 ~ t} = 0 Equation (2.32) now gives { 2 € Rd : r - 6 < \z - "AT7 - y\ < r + s} D S(-Y«) = 0 Since we were free to pick n arbitrarily large this proves the claim. Now equation (2.30) and the claim give OO 7~1 n=l so that for P6° — a.a.u 00 Xt(Rd \ p| AnnpT,.n"1)) = 0 (2.35) n=l It is possible to show that the map m — • m(R d \ H^Lj Ann(m,n - 1 )) is Borel measurable. So we can apply Theorem 1.9 and conclude that for any m G Mp{Rd) equation (2.35) holds for Qm — a.a.w.O 2.4 The range of the process over random time sets In Perkins [17] it is shown that with probability one the support process (S(Xt) : t > 0) has right continuous paths with left limits in the Hausdorff topology on compact sets. If we write S(Xt)- for lim,jt S(X,) then S(Xt)~ D S(Xt) and \Jt>0S(Xt)- \ S(Xt) is shown to be almost surely a countable set of points. We deduce that if A is a Borel subset of (0, 00) then (Jt€A S(Xt) is almost surely a Borel subset of Rd since for any to > 0 [j S(Xt)\ (J S(Xt)c\JS(Xt)-\S(Xt). teAn[t0,oo) t£An[t0,oo) «>o Note also that Hausdorff dimensions are unaffected by the addition of countably many points . In Dawson,Iscoe and Perkins [5] , if d > 4, suitable Hausdorff measure functions are found for the set Ua<t<bS(Xt) from which it follows that the Hausdorff dimension is 4. We now find the dimension 32 of \Jt£A S(Xt) for possibly random time sets A. This is an analogue of a Theorem of Kauffman for Brownian motion. In Kauffman [12] it is shown that if Bt is a two dimensional Brownian motion then with probability one dim(B t : t £ A) = 2dim(,4) for all Borel A C [0,oo) and the result is true for Brownian motion in dimensions d > 2. Notation. For R,K£ [0,oo) define rk = inf{i > 0 : Xt(Rd) = k) <rk = inf{< > 0 : Xt(B(0, k)c) > 0} Theorem 2.13 For d > 4 and any initial measure m £ Mp(Rd), with probability one dim ( |J S(Xt) 1 = 2 + 2dim(J4) for all nonempty Borel A C (0, r 0) (2.36) PROOF . Lower bound. Let Tk — k A A r t - i A ov We will show that if m £ MF(Rd) is of compact support, k £N satisfies m(R d) £ (Jb_1,ife) and e > 0 then for Qm — a.a.u dim ^ y S{Xt)^j > 2 + 2dim(^) - 2e for all nonempty Borel A C {k~l ,Tk) (2 37) If m £ M f ( R d ) is of compact support then almost surely the total mass remains bounded and the support remains bounded so that Tk ] T0 < oo as k —•co and dim(j4 l~l (k~l ,Tk)) T dim(A n (0, r0)). So we may take sequences kn | co ,e„ { 0 to conclude that if m £ Mp(Rd) is of compact support then the lower bound on the dimension of Ut£AS(Xt) in equation (2.36) holds Qm almost surely. For initial measures m £ MF(Rd) we can argue by decomposing m into countably many finite measures with compact support and use the branching property. Fix m £ Mp{Rd) of compact support, k £ {2n : n — 1,2,...} satisfying m(R d) £ (k~l,k) and e £ (0,1).Define a time grid as follows. Let = j2~n, T" = {tj : j = 1,2...}, If = [t?,t?+1). Along any fixed sample path we say 7" charges B(x,a) if there exists s £ J" such that X,(B(x,a)) > 0. For large n we do not expect many balls B(x,2~n) to be charged repeatedly by many of the Jj"'s. The following Lemma shows that there is not much mass in such balls . Lemma 2.14 For Qm — a.a.u and sufficiently large n Xt<x: Y l(lfnchargesB(x,2-n)) > 2nc) i < 2-( d + 5> n (2.38) { /?»c[*-',*] J 33 for all t 6 T3"+ 6n[0,ifc] For any fixed ball of radius a , if it is charged by the measure Xt we do not expect it to be charged much more than a 2 . In Perkins [16] (proof of Theorem 4.5) the following very precise result is shown. There exist constants c a,Ci > 0 such that for Qm — a.a.u and sufficiently large n Xt {x : Xt(B(x,2-n)) > cfllog(n)2-2"} < ck/n2 for all t E [k~l,k] (2.39) Define a space grid as follows . Let G£ = d~1^22~nZd 0 73(0, k + 1). Note that any point is in at most 3 d of the balls {B(xt,2-n) : x{ G G£). Lemma 2.15 For Qm — a.a.u and sufficiently large n sup Xt(B(x, 2"")) < 2^d+^nXt^(B(x,2"("-1))) (2.40) for all t G T3n n[0,ifc],x G G£. We delay the proofs of Lemmas 2.14 , 2.15 . The strategy of the proof is as follows . Given a cover C of \Jt£A S(Xt) we will construct a cover CA of A x [0, l/27fc]. Since A x [0, l/2Jfc] has dimension at least 1 + dim(yl) (see Falconer [9] Corollary 5.10 ) this will lead to a lower bound on how efficiently we can cover U(6y4 S(Xt). Fix u and n 0 < oo so that equations (2.38), (2.39) ,2.40 hold for n > n0 and so that zd2~n°22d+w < l/4fc and E n> n ocj/n 2 < l/4fc. Fix Borel A C (k~l,Tk) of dimension a. Since A C [0, crk) we have \JteA S(Xt) C 73(0, k). Choose a cover of \Ji€A S(Xt) C = {Bt = 73(x,,2-"') : i = 1,2...} with x,- e G^^m > n0 + 2. (2.41) We assume all the balls in C are distinct . Split the cover into two parts C\ = {73, • i E Ii},C2 = {73, : * € h} where S<ni-2) = | * : £ I(7/ ( n '- 2 ) charges 73(x, 2"("--2))) > 2<"'-2> > i e h if B{xit2-<ni-») C Sfni-3) i e h if 73(x,',2 _(n i _ 1^)n5(Cn._2) ̂  0 For each i G 72 find zt such that 73(s,-, 2-(n ,'_ 1)) C 73(zi,2-("i_2)) and less than 2( n ' _ 2 ) f of the intervals 7 2 ( n i - 2 ) g jfc_! ) f c j c h a r g e S(z.i2-(n.-2)) 34 Now we form the cover CA of A x [0,1/2Jfc] . Consider the balls in C 2 one by one in order of decreasing radius. For a ball B(xi,2~ni) G C2 there are less than 4.2( n i _ 2) £ of the intervals lfni C [k~\k] that charge J3(XJ,2~"<). For each J2"' that does charge S(x,-,2 _ n i) choose a rectangle in x R based above 72"' and of height c„ log(n,- — l )2 _ 2 ( n ' - 1 ) so that the base of the rectangle lies on top of any previous rectangle above I2"' ( or the x-axis if there are none ). Repeating this procedure for each ball Bi G Ci gives a collection of rectangles which we call CA- Note that each B(xi,2~"') gave rise to at most 4_2("i-2> rectangles of diameter less than (1 + ca)log(n,- - l)2- 2( n' _ 1>. We now check that CA covers A x [0,1/2k). Fix t G A and let {" = sup{<? : t] < t}. Let J„ = {i G h : tli = n) so that h = Un>n 0 +2 Jn- < 2^d+A> Y Xian(B(Xj,2-^-lS>) (by equation (2.40)) < 2(" + 4) n3 dX t- 3 n( (J B(Xj,2-("-V)) < 2<>d+VnZdXi3n I x : YI I(/ 2 ("" 2 ) charges S(x,2-(""2))) >2(""2)£ J3<"-3>C[Jt-i,Jfc] < 2 ( d + 4 ) n 3 d 2 - ( d + 5 ) ( n - 2 ) (by equation 2.38) So YMBi) = Y Eŵ -'2-")) «'€/i n > n 0 + 2 j e J „ < zd22d+102-n° < 1/4*. Since t < T t - i we know that Xt(Rd) > l/k. Let Ct = U ( „ > „ 0 + 2 ) { a : : Xt(B{x,2~n)) > ca log(n)2-2"} Equation (2.39) now gives 3/4fc < Xt(S(Xt)\Ct)' < Y Xt(Bi)I(BigCt,Xt(Bi)>0) i e/iU/a < l/4k+^2Xt{Bi)l(BigCt,Xt(Bi)>0) i e/3 35 If B(xit2-ni) £ Ct choose y< € B(xit 2~ni) n Cf. Then 1/2* < ^ ^ ( 5 ^ ' - 2 " ( n i " 1 ) ) ) I ( X ' ( S ' ) > 0 , 5 , g C t ) < ^ c6(ni - l )2 2 ( n i " 1 ) I ( [P , i , t 2 n i + 2~7ni) charges B.) (2.42) The right hand side of equation (2.42) is the total height of the rectangles in CA that lie above t. So indeed CA covers A x [0,1/2*]. Now suppose that dim([_L€j4 S(Xt)) < 2j < 4. Then we can find a sequence of covers {C m } m of the form 2.41 satisfying diam(Cm) —» 0 and Y | B , f 7 = ^ 2 " 2 7 n ' < 1 for all m Then diam(C^*) 0 and £ l#r+£ < ^ 4 . 2 < n i - a > ( ( l + c f l )(nj - i ) 2 - 2 ( « ' - 1 ) ) T + £ < Constant for all m. So Therefore and 1 + a < dim(A x [0,1/2*]) < 7 + e 27 > 2 + 2a - 2e dim( [ J S(X,)) > 2 + 2a - 2e and the lower bound is finished. Upper bound. The upper bound is a straightforward application of the global modulus of continuity for the motion of the particles in the nonstandard model for super Brownian motion. It will be enough to show that if m € Mjp(R d) and rn(R d) < * then for Qm — a.a.u. d i m ( y S{Xt)) < 2 + 2dim(,4) for all Borel A C (0, rk A * ) . t£A Fix rrif, G *M£(*R d) so that stjv/F(m/J) = m and fix * G N so that m(l) < *. Define for £, e_ G {j/f* • j G 'N} Hi, £) = {7 ~ t • 30 ~ t + £, 0 y 7, N?+L ? A} 36 Z(i,L) = \I(LL)\ Conditional on Ay Z(t,£_) has a *-Binomial distribution under *P with parameters fiNf(l) and pL = *P" ll°(N^(l) > 0). From Lemma 2.2 °(/ip£_) = 2 /£ whenever £ is nearstandard . Let Poisson(n) be a random variable having the Poisson distribution with mean n under Po. P (3i? € [0,/fc A rk] such that Z(t],2~n) > Jfc2(n+2)) < £ P [Z{t],2-n) > k2^n+2\rk > i?) «;e[o,t] < jfc2nP0(Poisson(/fc2n+1) > jfe2C"+2>) < Jfe2nexp(-ifc2n + 1(2-e1/2)) which sums over n. Thus for ui off a single null set we may find n0(w) < oo so that Z(t", 2 _ n ) < fc2("+2) for all t" £ [0, k A T*] and n > no, so that the global modulus of continuity for particles (Theorem 1.6a) holds with 2~n° < 6(u,3) and so that equation (1.12) holds. Fix u so that no(w) < oo. Fix A C (0,rk A k) of dimension a. Given £ > 0 we can find a cover C = {I?; : i = 1,2 ...} of A so that diam(C) < £ and \ p 2 " ' ( Q ' + e ) < 1 i Using the modulus of continuity we have U e U U e g ( J S ( / V 7 . i,3M2-("'-1))) (2.43) 7€/(<;;_ 1 ,2—o J i The right hand side of (2.43) forms a cover Cs of UttAS(Xt) of diameter less than 6h(2e) satisfying ^2 ]B\2a+2+3c < ^jfc2"'+ 2|5N/n72- n'/ 2 | 2 o + 2 + 3 e B e c s •' < Constant Choosing a sequence of covers of decreasing diameter gives dim((J.5(X t)) < 2 + 2dim(,4) forP-a.a.w. t£A It is possible to show that the set |w : for sufficiently large n, if 7" C [0,k A rk] then ( J t e / " ^(w<) c a n D e covered 37 by k2n+1 balls with rational centers and radius 3/i(2-(n~1))} is Borel in C([0, oo), Mp(Kd)). So by transfering to path space at the correct point in the proof we can show the lower bound holds Qm almost surely. • Notation. Define GMCn - |w : | A f - N[ I < Zh(t - for all nearstandard s,t,0~t satisfying 0 < t - s < 2 _ n , N[ £ A | Then from (1.15) P(GMC^) -+ 0 geometrically fast. Proof of lemma 2.15. It is enough to prove the lemma for the nonstandard model. P ( 3t G T 3 n D [0, Jk], x e GI s.t. sup X,(B(x, 2"")) > 2^+^nXt(B(x, 2"(n-1)))) \ «e [< ,<+2-»»] J < Y\ y " p ( G M C 3 „ n { sup X,(5(x,2-"))>2( d + 4)"X t(5(x,2-("- 1))})) t 6 T'-n (0 , t ]x€G ; V . e [ f ,«+2-»»] ) + P(GMCiJ (2.44) Now P ( G M C 3 „ n ( sup X,(B(x,2-")) > 2( d + 4 ) n ^(5(x,2-( n -^))) ) V «€[ t ,<+2-»») y < ' P I sup /i"1 Y W+. # A> € B^^-^"- 1))) V£€[0,2-»-] 7 „ T + > 2 ( d + 4 )" / i - 1 ^I(Af 7 e B(x,2-(n-V))\ 7~< / Let v = /i-'E^W € -B(x, 2~("-1))). Then using the *-Markov property at time t the process £T~«+, Wt^ A,TV7 G ^(x^-^- 1))) : £ > 0} has the same law as {A^(l) : s > 0} under P" . But P"( sup N?(l) > 2 ( d + 4) nA r 0"(l)) » € [ 0 , 2 - » » ] < P"(supX,(R d) > 2( d + 4 ) n X 0 (R d ) ) « > 0 _ 2-(d+4)" since X , ( R d ) is a continuous martingale. Substituting into (2.44) and noting the summation is over less than (k + l)d+12^d+3>" terms , Borel Cantelli gives the result. • 38 Lemma 2.16 For s £ T3n+6 0 [0, fc], p £ N there exists a constant C = C(p,k,m,d) such that for all " > (5 + loga(p))/e E" X,(x: I ( X t ? „ ( B ( x , c n 2 - n ) ) > 2 - 2 n ) > 2 n e ) t3ne[fc-l,ifc] < C n p ( d + 1 ) 2 - n t p . PROOF . Fix n and write Bx for £ ( x , c n 2 ") . C will be a constant depending on p,k,rn,d but indepen- dent of n whose value may change from line to line. X,(x: Y, l{Xtr{Bx)>2-2n)>2nt) tj-et*-',*] < E X,(x : Y, I ( * t 3 » ( S * ) > 2~2 n) > 2"£-2) ta"6[« + 2-:>»+',«:] +E X,{x: l(Xt,n(Bx)>2-2n)>2n€-2)\ (2.45) <3°e[ifc-1,»-2-3»+1] We will bound the first term on the right hand side of (2.45). It is similar and slightly easier to show the second term has the same bound. E < E X,(x : Yl I ( * t } » ( £ « ) > 2~2 n) > 2"£-2) * 3 "e[»+2 - 3 »+ i , i t ] / < C2-ncp E n Kxt?(B*) > 2 _ 2 n ) > 2 ( " £ - 3 ) p t » n «?»e| .+3-»'»+i,i .) >=i 31 Jp y m i n j , , i \t^-i^\>^~3n + i E E\J f[KXt3.?(Bx) > 2-2n)dX,(x) « : >"< . . . l :?»6 | . + 3 - 3 » + l,*| L «'=1 m i n k | « 3 n - t 3 . | > 2 - 3 n + i < C 2 - n t p 2 2 n p Y E \ f X t < ? n < . . . i 3 n e [ . + 3 - 3 n + l l h l £ y Xt>n{Bx). ..Xt,»(Bx)dXs(x) The following lemma gives an upper bound for such expectations. (2.46) Lemma 2.17 If fc-1 < s < < t 2 < . . . < tp < k then E Xu(B{x,a))...Xtp(B{x,a))dX,(x) where C depends on k, d, m,p but not a or the ti's < Capd[(r„ - ! „_ ! ) . . . (i 2 - tOC*! - S)}1-"'2 39 Proof of Lemma 2.17 . The required expectation is the standard part of _ _ _ ̂  p or,~ti or,,_1~t,_i xv-1 £ • = K(« p-«,-i) (2.47) ^ - ( p + 1 ) £ E - E n ^ - ^ ' i ^ ) «=i n w - ^ - - i<o) L l = l E 'Eim?; - N?\ < cT(ap:/3,oi o,_,)=i where ir(a : 7 1 , . . . , 7 n ) = |ar| — inf(ji : a\j ^ 7,|j Vi = 1,..., n) is the number of generations back that a branched off from any of the branches 7 1 , . . . , -y„. Recall Y(t) is a *-Brownian motion under 'Po- If i G {n(tp — tp-i), • • • ,ptp] then E 'E[i(\Kr ~N-\^ a ) i -V°i «,-x)] (7(a,:/3,Oi,...,orp_i)=t ' p'P0(\Y(iii-i)\ < a) So < ^ E if i ^ /itp • = M(<F-'I.-I) (/xm"(-Rd) - 1)*P 0(|Y(« + tP)\ < a) if i = /it, E ^ [ K K ' - i V f i ^ ) ^ , , « , . , ) ] » ( o , : » 9 , a I , . . . , a F _ i ) = i < c(m(RV(«+<pr d / a+"V 1 E ( » > _ 1 ) " d / 2 < C a ^ - t p . j ) 1 - ^ 2 . Substituting into (2.47) and using induction over p gives the result Completion of proof of Lemma 2.16. Using Lemma 2.17 and the bound in equation (2.46) we have X.(x: E I ( ^ < f (B.) > 2~2 n) > 2— 2) *3"£{*+2-3n+1,Jfc] < C n ^ - d - ^ " E -t%Wu-»))1-d,a < 3 "<. . . t 3 »6I«+3 - a n + 1 .*) " 1 j k + i ik '- < Cn'ty*-'-'*' [" dti ̂  dt2... fk dtp [(tp - <p_0 . . . (t3 - <!)(*! - s)]1-"'2 < C n ( p + 1 ) d 2 ~ ' " p . 40 Notation. Define for j,n G N, x G *R the events A?(x) = {3 T ~ (j - l ) 2 - 2 n s.t. /V ( 7_ 1 ) 2_ 3„ G £ ( £ > 9 M 2 - 2 " ) ) , / V j 2 - 3 n ( 7 ) > 0) = [3y ~ (j - 1)2"2" s . t . ^ _ 1 ) 2 _ 3 „ G 5( £ >9/i(2- 2")), yVi2-a„(T) > 2"2"} A](x) = {3s G //n,7 ~ s such that "A7 G 5(2,12/i(2-2"))} 75;(x) = {X i 2- 3n(5(x,12ft(2- 2")))>2- 2"} Note that up to a null set Lemma 2.16 gives a bound on E A](x)DGMC2n C A?(x) B?(z)nGMC2n C Bfix) t 2 3 n ^ i=Jfc-'23» (2.48) (2.49) (2.50) whereas to prove Lemma 2.14 we wish to bound * 2 3 n (2.51) By restricting to GMC2n equations (2.48), (2.49) will allow us to replace A^J{x),B^(x) by AJ{x), B?(x). We will show that each time A"(x) occurs there is a good chance that B"(x) occurs and use the following Lemma to convert our bound on (2.50) into a bound on (2.51). Lemma 2.18 On a probability space (fl,(•? r;) j ez + ' p) let An,Bn,n = 1,... ,N be events satisfying for some q G [0,1] i. Aj,Bj efj. ii. Bj C Aj. in. P ( S J | J ; _ 1 ) > 9 ^ 1 ^ - 0 . Then (TV TV \ 32*Ai ^n,32lBi < a\?o < Po(B(n,q) < a) P - a.s. i=i «=i / where B(n,q) has a Binomial distribution under P0 with parameters n,q. 41 PROOF. Define r 0 = 0, r,- = inf(m > T j _ ! : u G Am) for j — 1,..., N. Let = 1^, Yj = Ia,.We claim E\YTjl{Tj < oo)\TTj_x] > qE{l(Tj < co)|^ T i_J for j = 1,..., AT. (2.52) To prove this pick C G -Note that { r j = n } = | J yi„n {(*!,..., *„_!) = *)} and this union is disjoint. If x G {0, l}" - 1 satisfies £ x < — 3~Itnen {(-^l, • • • ,-^n-i) = z} Q {Tj-i — ̂ } for some fc = 1,..., n — l , s o that c n {(x, , . . . , * „ _ ! ) = x} = c n {(Xi,...,*„_!) = x} n{r,-^ = fc} G ̂ „_ I ; Then / yT ii(Tj < oo)dP = J2 I Y » d p Jc n = 1 Jcn{Ti=n} = E E / = E E / *w > ?E E / n = l l 6 ( M ) . - . - ' C n ( ( X 1 , . . . , X n . 1 ) = I } = 1 J * T , K > j < 00)dP = 1J I(TJ < co)dP /c which proves the claim. We now check by induction on n that for n = 1,..., N , a = 0,..., n P Ê Y ' i ^ a' r" < j ̂  Po(B(n,q) < a) (2.53) The case n — 1 is immediate from (2.52). Assume equation (2.53) for n = 1,..., fc. k+1 P(52YTi<a,Tk+l<oo\rTi) 42 = ! ( E y ^ = a)P(Yr>+l = 0,^+i < oo|J-TJ + I ( E ^ < a - l ) P ( T k + i < oo\FTk) i=i < (1 - ? ) I ( E Y T i = a) + I(£y T j. < a - 1) P ( r t + 1 < o o ^ J i=i k i=i < (a - ? ) l £ ^ < a) + ? I ( E ^ < a - 1) j I( Tk < oo) since {rk < oo} C {rk+i < oo}. So taking conditional expectations given To and using the induction hypothesis k + l P(Y,YTj<a,n+i<oo\T0) < (l-q)Po{B(k,q)<a) + qP0{B(k,q)<a-l) i=i = P0(B(k+ l,q) < a) completing the induction. Finally • ff N N P(J2lA,>n>J2lB,^a\:Fo) = P(rn < o o , < a \ F 0 ) i=i i=i i=i < F ( r n < o o , E ^ < a|^0) i=i < P0(B(n,q) < a). Proof of Lemma 2.14. Fix n G N and nearstandard t G {j/n : j G °N},/3 ~ £ such that /vf ^ A . Let j'"(t) = sup{j : j2~n < t}. We will apply Lemma 2.18 to the events A"(Nf), BJ{NP), j = k~l22n,..., k22n, j ± ]2n(t) + 1 and the internal filtration = Aj2-*~ V : s. < t) j = k-^2",.. .,k22n, j ± j2n(t) + 1 under the internal probability 'P. Conditions i. and ii. of the Lemma is immediate and we claim that condition iii. holds , namely that there exists g G *R d satisfying q = exp - 2 such that for j ^ j2n(t) + 1 •P(BJ*(/Vf )\QU) > I'PiA^Nl)^) (2.54) Before proving (2.54) we complete the proof of Lemma 2.14. For t G T 3 "+ 6 fl [0, k] E * 2 2 Xt{x: E l(^(x))>2nc}l(GMC2n) ;=i t-'2 3" 43 < E * 2 3 Xt{x: £ l(B?(x)) > 2n<'2} (2.55) * 2 3 " * 2 3 " +£ E l(A?{x))>2*", £ I ( B ; n W ) < 2 f l t / 2 } I ( G M C 2 „ ) ; = J t - 1 2 a » j = i - i 2 3 » Lemma 2.16 gives the upper bound Cnf(d+1h-ncp/2 for the first term in (2.55). The value of the second term in (2.55) is less than the standard part of But by Lemma 2.18 / * 2 3 Jfc23 E ^ f ) ) ^ ' 1 - E I ( ^ ( A r f ) ) < 2 n f / 2 , G M C 2 n > 7 V f * 2 3 * 2 3 E I ( ^ r , ( 7 V f ) ) > 2 " r - 1 ) E l{B»(N?))<2n*l\GMC2n \Qnk. Wj3n(i)+i jVj ' 3"( i)+i < • P 0 ( y ( 2 n £ - 1 , o ) < 2 n e / 2 ) So (using (2.48)(2.49)) the second term of (2.55) is bounded by P~l J2'P(N? * A ) * p o ( y ( 2 n t - 1 , 9 ) < 2 n E / 2 ) < m ( R d ) ( l - e " 2 ) 2 " " for large n So Xt{x : E c h a r S e s B(x,2"")) > 2n<} > 2^d+h> for some t £ T3n+6 n [0,*]} /?"C[t-»,t] < E 2 ( r f + 5 ) " £ ; teT s "+ s n[o , * ] Jfc23 Xt{x: E I(i"W) > 2 " £ } I ( G M C 2 n ) < C23n+62<d+Vn [ n(<*+i)P2- n tP/2 + ( 1 _ e - 2 ) 2 " / ' l + P r G M C C n ) + P(GMC<2n) for large n.Taking p — 2 ( d + 9)/e , Borel Cantelli gives the desired result. I t remains to prove (2.54).Fix n,j,t,0~ t such that t & [(j - l)2~2n, j2~2n]. For j ~ (j - l)n2~2n define . A?(y,x) = {N^_1)2^eB(x,9h(2-2n)),Nj2-3n(y)>0} 2*7(7,*) = { ^ _ 1 ) 2 - 3 » G B ( £ , 9 / l ( 2 - 2 n ) ) ! ^ 2 - 3 n ( 7 ) > 2 - 2 n } 44 Then AW = U A?b>ti 7~(;-«)2- 3 " = (J s;(7,£) Fix 5 = (71,... ,7r) where 7l- ~ (j — l )2 - 2 ",r > 1. Consider the set r Asia) = n '̂̂ .aO n f l W?.*)) e (2-56) t = l , « ( / - l ) 3 - ' * 7^5 As S ranges over nonempty subsets of {7 : 7 ~ (j — 1)2-2"} the sets As(x) form a disjoint partition of A^x). 'P (^B"(Nf)\Q"_l, As(Np)j > 'P (/V;2-2n(7i) > 2-2n\g^lyAs(N[)) > 'P (7V j a - a .(7i) > 2 - 2 " | / V J . 2 - 3 . ( 7 l ) > 0) l(As(N[))I(/%_1)2-3„ ^ 7 l ) + ^(7^2 -3 . ( /? | 0 - - i )2 -»- ) > 2- 2 n|/V J-2— ( % - i) 2-*0 > o./vf # A) x I ( ^ s ( 7 V f ) ) I ( / ? | 0 _ 1 ) 2 - 3 „ = 7 1 ) Lemma 2.2 b. c. gives •P(7V j 2 -3»(7i) > 2- 2 "|7V i 2 - 3 »(7i) > 0) « e"2 •P(7v J-2-3»(/?|(j_i)2-a» ) > 2 - 2 " | 7 V - J . 2 - 3 „ ( % _ 1 ) 2 - 3 „ ) > 0,/vf # A) = •P ( 7 V i 2 - 3 . ( / 3 | ( j _ 1 ) 2 - 3 . ) > 2- 2 "|7V/ 2 _ a „ ± A) « 2e- 2 So with £ « e - 2 'P{B?(Nl)\g?_ltAs{Nl)) > ql(As(N[)). Since this is true for all sets As(Nf) of the form (2.56) we have and the proof is complete.• 45 Chapter 3 The martingale problem characterisation 3.1 The measure of a half space The martingale problem satisfied by a superprocess gives a semimartingale decomposition for Xt(f) where / is in the domain of the generator A of the underlying spatial motion, namely Mf) = "»(/)+ f X,(Af)ds + Mt(f) (3.57) Jo (M(f))t = f X,{f2)ds Jo We look for a similar decomposition of Xt(f) for general bounded measurable / . Perkins has shown (private communication) that if the semigroup of the underlying process satisfies a continuity condition, for instance 3C,/3i,/32 > 0 such that for all 0 < 6 < u, f G bS \\Tv+if - Tyf\\ < C\\f\\ (i/-*5* V 1) (3.58) then with probability one the processes t —• Xt(f) for bounded measurable / are all continuous on (0,oo). The proof shows that if /„ G D(A) are uniformly bounded and converge pointwise to / then almost surely the paths Xt(fn) converge to Xt(f) uniformly on compact subintervals of (0,co). Also " i ( / n ) —* m(f) by dominated convergence and £(sup(M t(/„) - M,(/m)) 2) < 4E( [T X,((fn - fm)2)ds) - 0 (3.59) t<T Jo as rn,n —• oo again by dominated convergence. So along a subsequence n' the martingales Mt(fn') converge almost surely and uniformly on compacts to a continuous martingale Mt(f) . So under the hypothesis (3.58) ,with probability one the processes f* X,(Afn)ds have a subsequence which converges uniformly on compacts in (0,co) to a continuous limit. We examine the case where Xt is a super symmetric a— stable process (so that hypothesis (3.58) is satisfied with 0\ — = 1) and / is the indicator of a halfspace . Theorem 3.2 shows that Xt(f) fails to 46 be a semimartingale i f 1 < a < 2. We will need the existence of a density for Xt in dimension 1 when 1 < a < 2. We state the necessary results as a Theorem. T h e o r e m 3.1 Let m £ Mp(R) have a continuous density u(x).Let a £ (1,2] and Xt be a one dimen- sional super symmetric a—stable process starting at m defined on a probabilty space (Q,F,P). Then Xt has a density X(t,x) which is continuous on [0,co) x R. There is a space-time white noise WtiX defined on an enlargement of(Q,Jr, P) such that for all f £ C°°(R) of compact support Xt(f) = m(f)+ f X,(Af)ds+ I f y/X(s,x)f(x)dWSiX Vt > 0 (3.60) Jo Jo JR For fixed x £ R,t > 0 X(t,x)=Ttu(x)+ I j Pt-,(y,x)^X(s,y)dW,iy (3.61) Jo JR where Pt{x,y) is the a—stable transition density. If u(x) is bounded and uniformly Holder continuous then there exist j > 0 and C depending only on m and a such that E((X(t,x) - X(s,x))2) < C(t - Sy for allt,s>0 (3.62) The existence of a jointly continuous density satisfying (3.60) is proved in Konno and Shiga [14] Theorem 1.4. Equation (3.61) is established during the proof in Konno and Shiga (although they consider more general init ial measures and thus work on [<o,oo) for to > 0 , it is easy to extend (3.61) to [0, oo) for init ial measures that have a continuous density.) The proof uses moment estimates of the type in (3.62) but since we can't point to exactly what we need we give a proof. PROOF OF (3.62) . From (3.61) X(t,x)-X(s,x) = (Tt-T,)u(x) + JQ J(Pt-r(x, y) - P.-r{x, y)WX{r, y)dWr,y Find C,0 £ (0,1] such that \u(x) - u(y)\ < C\x - yf for all x,y £ R. Then \\(Tt-T,)u\\ < \\(Tt.s-I)u\\ < CE0(\Yt-sf) < C{t-sfla 47 The stable density satisfies p i (x ) < C(|2:|~ ( 1 + a) A 1) and the scaling equation pt(x) = t~1/ap1(t-1^ax). £ ( ( j f Pt_r(x,yWX(r,y)dWt,y)2) = J* mTr(p2_r(x - -))dr < \\u\\ / / p2(x)dxdr J0 Joo / pt — s < C r-lladr Jo = C(t-s)^a-^'a Similarly E(( f ( P t _ r - p,_r)^/x(^)dWr,y)2) < C{t - s ) ( ° - 1 ) / 2 a . J,-(t-,y/3 -('-• ) I / 3 Finally \\pt -p,\\ < C(t - s ) s - ( 0 + 1 ) / ° for 0 < s < t so E(( f (pt_r(x, y) - p,_r(x, y))y/xJr~^dWr>y)2) Jo < Cm(l) /' (t - s)2r-2(a+1Vadr < C(t-s)WW"\ o Theorem 3.2 Let m G Mp and Xt be a super symmetric stable process of index a. Let H be the indicator of a halfspace .Define (j>(a) = 2a/(a + 1) if a > 1 Then for any T > 0 we have the following decomposition. Xt(H) = m(H) + Vt +Mt forO<t <T where Mt is a continuous L2 martingale satisfying (M)t = / * X,(H)ds and Vt is continuous on (0,T]. If 0 < a < 1 and m has a bounded density then Xt(H) is a semimariingale and Vt has integrable variation on [0,T]. If 1 < a < 2 and m has a bounded density then Vt has integrable <j>(a) variation on [0,T\. If in addition the density is uniformly Holder continuous and satisfies u(0, x) > 0 for some x on the boundary of the halfspace then with probability one V< has strictly positive <j>(a) variation on [0,T] and hence Xt fails to be a semimariingale. 48 The proof uses the following well known Green's function representation for Xt{4>) ,<f> G bS. Xt{4) = m{Tt<j>) + f I Tt-,4>(x)dZ,,t (3.63) JO JE where Tt is the semigroup of the underlying motion and Zt>x is an orthogonal martingale measure satisfying (f I f{s,x)dZ,tX) = f X,{f2{sr))ds (3.64) Jo JE JO for any measurable f(s,x) such that E{JQ X,{f2{s,-))ds) < oo, Vi. For the theory of stochastic inte- gration with respect to martingale measures see Walsh [24].Briefly, equation (3.63) may be derived from the martingale problem as follows. Rewrite (M) as Xt(f) = m(f)+ f X,{Af)ds + f f f(x)dZ,,x. Jo JO JE Considering functions of the form ft(x) = ^5i(x)/i,(i) and then passing to the limit we have Xt(ft) = «»(/o) + /' X.(Af. + df,/ds)ds + f f f.(x)dZ.,, (3.65) Jo Jo JE for sufficiently smooth f,{x). Fixing t > 0 and checking that f,(x) — Tt-,4>{x) is smooth enough to apply (3.65) we immediately obtain (3.63) for nice <j>. Extension to all <f> G b£ is straightforward. PROOF OF PROPOSITION 3.2. It is enough to consider the case d = 1 and H = I(x > 0). We start with the Green's function representation. Xt{H) = m(TtH) + f f Tt.,HdZ,iX Jo JR If we set Mt = f* f HdZ,iX then Mt is a continuous L2 martingale satisfying (M)t = /0* X,(H)ds. The decomposition follows by setting Vt = m((Tt - I)H) + f j {Tt-, - I)HdZ,iX. Jo JR Now m((T, - I)H) = m{{Tt - I){H)l{x > 0)) - m((7 - Tt){H)l{x < 0)) is the difference of two decreasing processes and so of bounded variation. It remains to check the variation of Wt:= f I (Tt..-I)HdZtit. (3.66) Jo JR An upper bound for the expected value of the size of an increment of Wt can be obtained using the isometry for Z, x (equation 3.64 ) . We delay the calculations and state the result as a Lemma. 49 L e m m a 3.3 If m has a bounded density then there is a constant C depending only on T,a,m such that /or 0 <s <t <T 2 ^ ̂  I ( < - s ) ( a + 1 ) / a ifa> 1 E((Wt - W,) ) < C (t-s)2 ifa<\ Since we are interested in a continuous version of Wt it is enough to check the variation over one sequence of decreasing nested partitions .Let A = T/n and Sj = jA . I f 1 < a then E^plW^-W.^*^ < YmW^-W,^)2))^'2 < CT. So Wt and hence Vt has integrable <f>(a) variation on [0,T]. Similarly i f a < 1 then Vt has integrable variation on [0, T\. We now assume that u(0, x) is bounded, uniformly Holder continuous and satisfies u(0,0) > 0. I f 1 < a < 2 then Xt has a jointly continuous density u(t,x) and / / f(s,x)dZ,iX= [ f f(s,xWu(s,x)dWs,x Jo JR Jo JR where W,tX is a space-time white noise (see Theorem 3.1). We split an increment of Wt into three parts as follows. Fix n and let tj = j/n . m i + l - W t i .= f ' [ (Tti+l-.-Tti..)HdZ.lT Jo JR + £+lJR(Tti+l.t-I){H)y/^^dWtttl + j*'** JR(T<>+>- - I)(H)(V^)- yf(t~x~))dW,,x = • Cj + Cj + We wish to show that Wt has strictly positive <j>(a) variation . We will first show that \^a^> is small and does not contribute to the variation. Then noting that Q is Tti measurable , we will show that conditional on Tt} , €j has a mean zero Normal distribution with variance more than Cn-1Xtj(B(0,n-1/a)). since Xt has a density u(t,x) bounded away from zero at t = x = 0, this variance will be of the order of n-(o+i)/or a n d t h e i n c r e m e n t |£,- + Cj|* ( o ) will be of the order of n _ 1 . E(\ V\2) < E ( £ + 1 JR(Tti+1-. -7)2(//)(\AM- y/u(tjtx))2dsdx) 50 (T, i + 1_. - I)2(H)[E(u(s,x) - uitj.zWdadz < Cn-i R •A, JR (Ttj+l-, — I)2(H)dsdx where 7 > 0 from Lemma 3.1 which uses the Holder continuity of u(0, x). Using the bound (Tr — I)H(x) < Cr\x\~a A 1 ( see equation 3.71) we have / (Ttj+1-,-I)2(H)dsdx < C n " 1 / dx+ / n-2|x|" • / i , J R WO Jn-i/o 2a dx So 75 E \ * \ " a ) >=0 < C n ~ ( a + 1 ) / o . [ n T ] - l < E ( ^ ( v j ) ) * ( a ) / 2 j = 0 n [ T ] - l < C E (n-(0 + 1>/"n-Y(°:>/2 = C n ~ 7 a / ( ° + 1 > . Conditional on T%i , €j has a Normal mean zero distribution with variance (T, i + 1_, - 7)277ds Let YT be a symmetric a-stable process under PQ . \(Tr-I)H(x)\ = P 0(Yi > Ixl/r1/") > W i >2 1 / a )I( |x |<(2r) 1 / a ) So f3+l(Ttj+^,-I)2Hds > ^ (P0(Y1>21>a))2l(\x\<(n)-1'")ds = C2n-1I(|a:| < n" 1 / a ) where C 2 = (P0(Yi > 2 1 / ° ) 2 / 2 . Let N have a Normal mean zero variance one distribution under Po- Q(|f; l* ( o ) > Kn-l\?tj) > Po(N2 > CilKil*0W-W°)-1)/xt.(B{Q,n-lla))) > (1/5)1 \xti(B(0, n"1/")) > C2-lK2l^n-lla (3.67) 51 using P0(N2 > 1) > 1/5. Since Cj is T%i measurable Q(Vj + C i l * ( o ) > Kn-1^) > (1/10)1 [XtjWOtn- 1'")) > C 3 _ 1 K 3 / *( 0 5n- 1 / o ] . The density u(i,x) is jointly continuous and u(0,0) > 0 so given e > Owe may find no > 1,K0 > 0,i 0 > 0 so that for all n > no Q(Xt(B(0,n-l'a) < C^Ka'^n-1!" for some 0 < t < t0) < e. Then for n > no [nT]-l Q( E k;+0l* ( o )>'co<o/20) [nT]-l > Q( E I ( k i + O I * ( t t ) > W « ) > " W 2 0 ) j=o /[nT]-l > Q I E 1(^(5(0,n" 1/")) > C^J^n-1'0) > nt0 /[nT]-l [nT]-l E I ( k > + 0 l * ( a ) > « o / n ) < n < o / 2 0 j=o > ( l - e ) -P 0 (S (nto , l /10 ) < nto/20) where B has a Binomial distribution under Po, using Lemma 2.18. So for large n Q( E l c i + <i l * ( 0 ) ^ «o<o/20) > 1 - 2e. 3 = 0 But from (3.67) for large n n[T]-l <?( E h\HQ)> «o«o/40) < £ . j=0 Now Minkowski's inequality and Fatou's Lemma give n|T]-i <3( E l^' i+i ~ > «o<o/80 infinitely often ) > 1 - 3e. i=o Since e was arbitrary it follows that the <j>(a) variation of Wt over [0,T] is strictly positive. • 52 For fixed x > 0 PROOF OF L E M M A 3.3 . From (3.66) E[(Wt-W,)2] = £? (T,_P - l)HdZr,x + JVt-r " T,-r)H dZr<x)2 = J mTr((Tt-r - I)H)2)dr + J* mTr(((Tt.r - T,-r)H)2)dr. (3.68) (Tr+6-Tr)H(x) = P0(Yr+6 > -x) - P0(Yr > -x) = Po (y 1 €[ Z / (r + 6 ) 1 / ° ) x / r 1 / t t ] ) < Kx/r 1 /") - fx/(r + «)1/a)|pi(x/(r + /5)1/0). C will be a constant depending only on T, a, m whose value may change from line to line.Using the bound pi(x) < C( |x | - ( 1 + a ) A 1) we have for r > 6 \(Tr+6-Tr)H(x)\ < C(6\x\-aA6\x\r-Wa) < C6(\x\-° Ar-1) (3.69) for r < 6 \(Tr+s-Tr)H(x)\ < P 0(yi€[|*|/* 1 / O,oo)) < C6\x\~a Al (3.70) and for r > 0 \(Tr-i)H(x)\ < p0(netkl/»-1 /o,oo)) < Cr\x\—Al (3.71) Find a constant K so that the densities of the measures mTr are bounded by K for all r > 0. From (3.69), for 0 < r < s - (r - s) mTr((T,.r - T,.r)2)H ( r(-ryl" fl \ < 2(t - s)2 [K(s - r)~2 / dx + K \x\~2adx + mTr(\x\ > 1) \ J0 7(a-r)i/» J 2 . l + is-r)^'^-2 if a #1/2 < C(t- s) I l+log+ ^ s - r ) - 1 ) if a = 1/2 53 From ( 3.70), for s - (t - s) < r < s m T r ( ( T t _ r - T.-T)2)H At—)1" < 2CK dx + 2CK(t-s)2 \x\-2adx + 2C(t-s)2mTr(\x\>l) Jo J(t->y/a (< _ s ) ( 2 A ( l / a ) ) i f a j . 1 / 2 < C {i-s)2(l + \og+((t-s)-1) if a = 1 / 2 So m T r ( ( T , _ r - T,-r)2)Hdr («-') /•'-(<-•) [ 1 + ( 5 _ r ) ( l / a ) - 2 Jo fit-,) f,-(t < C(t - s)2 / dr + C(t - s)2 Jo J(t~,) •r j.-(t-,) +C ' ' ' (t-s)2(l + log+((t-s)-i)) < \ (t-s)2 i f a < l \ (t - s ) («+ 1 ) /« if a > 1 Similar arguments give an upper bound of no larger order for the first term in (3.68). • Remarks. i. For any m £ Mp(Rd) similar arguments show that if 0 < S < T, 0 < a < 1 then Vt has integrable variation on [S, T}. i i . I f 1 < a < 2 then the instantaneous propagation of the support (see (3.78) ) implies that Vt will have strictly positive <f>(a) variation on [0,T] for any T > 0. I f a = 2 and m ^ 0 then there is positive probability that for some s > 0 the measure X, wil l have a uniformly Holder continuous bounded density that is strictly positive at some point on the boundary of the halfspace. Thus for any XQ ^ 0 the process Xt fails to be a semimartingale. i i i . Sugitani [23] shows that for super Brownian motion in dimension one the local time process Y(t, x) = X(s, x)ds is differentiable in x and that i f m is atomless the derivative DxY(t, x) is jointly continuous in t,x almost surely. We can easily identify the drift term Vt in the decomposition of Xt(H) as {\/2)DxY(t,x) . Take m 6 A f f ( R ) atomless and of compact support. Define / a ( x ) = ((x — a) V 0 ) 2 . We may find / „ G D(A) so that / „ | fa{x) and Af„ —• I(x > a) bounded pointwise. We have enough domination 54 (e.g. E(supt<T Xt(f%)) < oo ) to take limits in the martingale problem and obtain Xt(fa) = m ( / 0 ) + / X,(l(x<a))ds + Mt(fa) Jo = rn(fa)+ f Y(t,x)dx + Mt(fa) (3.72) J a We wish to differentiate (3.72) twice with respect to a and again we have enough domination. Thus for a fixed t 2Xt((x - a) V 0) = 2m((x - a) V 0) + Y(t, a) + Mt(2(x - a) V 0) (3.73) Now continuity of both sides in t gives (3.73) for all t. Repeating the argument and using the continuity of DxY(t, x) gives Xt(l(x < a)) = m( I (x < a)) + (l/2)DxY(t,a) + Mt(l(x < a)). 3.2 T h e death p o i n t We use the characterisation of a Superprocess Xt as a solution to a martingale problem (equation 3.57) to study the sample path behaviour near the time of death. Set £ = i n f > 0 : -Xt ( l ) — 0} where we write 1 for the constant function with value one. I f m 6 Mp then f < oo almost surely. Define Ct= ( l/X,{\)ds. Jo In Konno-Shiga [14] Theorem 2.1 it is shown that with probability one Ct is a homeomorphism between [0,£) and [0,oo). Let Dt : [0,oo) —• [0,£) be the continuous strictly increasing inverse to C ( . Shiga [22] uses Dt as a time change together with a renormalisation to convert a class of measure valued processes into a class of probability valued processes. The Superprocesses studied here do not seem to fall directly into his context. However the time change will still be useful. By stretching out the interval [0,£) into [0, oo) we can use the behaviour at infinity of the time changed process to give infomation about Xt before death. For t G [0,oo) define Yt = XDt Yt = Yt/Y,(l) Gt = T̂>, 55 Note that {Yt : t > 0} is a probability valued process.We derive the martingale problem for Yt. For / € D(A) rD, Yt{f) = rn(f)+ j ' X,(Af)ds + MDl(f) Jo = m(f)+ ff.iA^Y.i^ds + Ntif) Jo 10 where, since Dt is a continuous time change , Ntif) is a continuous Qt local martingale satisfying (N(f))t = ( 'x,(f2)ds Jo = f Y,(f)Y,(l)d. Jo In particular Y t(l) = m(l) + A>((l) (N(l))t = f{Y,{l))2ds Jo (N(f),N(l))t = fY,(f)Y,(l)ds Jo Applying Ito's formula and noting that Y«(l) > 0 for all t > 0 we have Yt{f) = mif) + fY.iAf)ds + Ntif) (3.74) Jo where Ntif) is a continuous Gt local martingale satisfying (N(f))t= fY.if2)-iY,(f))2ds Jo The martingale problem for Yt is frustratingly close to that for the probability valued diffusion known as the Fleming-Viot process ( where the drift term in (3.74) would be replaced by fY,iAf)ds ). In Konno-Shiga [14] this 'connection' between the martingale problems is used to derive the existence of a continuous density for the Fleming-Viot process in dimension 1 from that for super Brownian motion . The following result shows that as t —* £ , what mass that remains is concentrated near a single point. Theorem 3.4 For m € MF there exists an E valued random variable F such that with probability one Xt/Xtil)->6F ast — £ (3.75) 56 where the convergence is weak convergence of measures. The law of F given the history of the total mass process Ti = a(Xt(l) : t > 0) satisfies Em(f(F)\H)=l/m(l) f T6fdm. (3.76) JE Remarks. i. Equation (3.76) implies that the law of F can be constructed as follows. Position a particle in E at random according to the measure m ( ) / m ( l ) . Let the particle move according to the underlying spatial motion but independently to the process. Stop the particle at time £. The final position of the particle wil l have law F. i i . The law of £ is given by P(£ <t) = e x p ( - 2 m ( l ) / t ) PROOF. First assume E is compact. Take / £ D(A). fY.{Af)ds < \\Af\\ frt{\)ds Jo Jo = \\Af\\ f XD,(\)d8 Jo = \\Af\\Dt < \\AfU. So Nt(f) > -m(f) - \\Af\\Z. For any continuous local martingale (Mt;t > 0) , with probability one either Mt converges to a finite l imit or l i m s u p M , = — l imin f M ( = oo (see Rogers Williams [20] Corollary IV.34.13 ). So Nt(f) converges as t —+ oo to a finite l imit. Also \f. Yr(Af)dr < \\Af\\(Dt - D,) — 0 as s,t — oo. So f* Y,(Af)ds converges as t —+ oo.Thus Yt(f) converges a.s. to a finite l imit which we call Voo(/)- Since C(E) is separable and D(A) is dense in C(E) we may pick {<£„}„ C D(A) dense in C(E). Off a null set N we have Yt(<t>n) —* Voo^n),Vn. Fix w £ N. Then by approximation Yt(f) converges to a finite l imit Yoo(/) for all / £ C(E). Also / —+ Yco(/) is a positive linear functional with Voo(l) = 1 and thus arises from a probability which we call Yoo. For / £ D(A) A ? ( / ) - fY.{f2)-{Yt{f)?ds Jo is a continuous local martingale. Since Nt(f),Y,(f2),Y,(f) all converge to finite limits this local martin- gale must converge requiring Yoo(/2) = (Yoo(/))2 a.s. So the probability is concentrated on a level 57 set of f . But E is a metric space so that C(E) and hence {4>n}n separate points and this forces Yco = 8p a.s. for some F. We have been unable to deduce the law of F directly from the martingale problem but it comes immediately from the particle picture. Take the nonstandard model with — fi~r ^ 6Xi satisfying sijKpfffl,,) = m. Let Q be the internal algebra generated by the total mass process {Nt(l) : t > 0} and a(G) the standard <7-algebra generated by Q. Note that NA['E) is Q measurable for all t so that 7i is a sub cr - algebra of cr(G). Let £„ = inf(t > 0 : N£(l) < 1/n). For any / € C(E), n > m ( l ) - 1 •E(N?J-f)/N?n(l)\G) = n'E(n-> £ 'f(N&\G) = n/x- 1 £ 'P(Nl ? A\G)Tuf(7\o) Now £ = 'P(N^n ^ A\G) is independent of j, so -E{N*mCf)/N{n(l)\G) = n ^ ^ ^ E ^ / f x , ) ) i = n / m ( l ) / ^ / ( x ^ m ^ x ) ^ - 1 E ^(AT f 7 n ^A|a)) E f~( n = l / m ( l ) / 7>„/(fm„. So f5(X. (f)/X (l)\<r(£)) = l /m( l ) / T 0 / d m using Albevario et al. [1] Proposition 3.2.12.Now °£„ f £ as n —• oo so that E{f{F)\H) = l / m ( l ) / T £ / d m . (3.77) When E is only locally compact we can extend the semigroup Tt to E U {oo} the one point com- pactification of E by taking T t (co, {oo}) = l ,T t ( x , {co}) = 0 for all x G E,t > 0. Working with this new Feller process on E U {oo} the above argument gives the existence of a death point F taking values in E U {co} and satisfying (3.75) and (3.76). Since P(£ < co) = 1 the characterisation of the law of F (3.77) ensures P(F € E) = 1. • Example. Let Xt be a super Poisson process. Define 7 i = inf(< > 0 : Xt({0,..., k — 1}) = 0). In 58 Perkins [17] Corollory 3.1 it is shown that Tk t £ and S, = {k,k+ 1,...} for Lebesgue a.a.t in [Tk,Tk+1),k G Z + , P m - a.s. Theorem 3.4 shows that only finitely many of the Tk's are distinct. Indeed 0 = To < Ti < . . . < 7> < 7>+i = TF+2 = . . . = /; P m - a.s. There is positive probability for any combination of equalities amoung To,T\,... ,TF. 3.3 The support near extinction The closed support of a superprocess Xt at a fixed time has been studied in Perkins [17] and Evans and Perkins [8] . If the spatial motion is a Levy process on R d with Levy measure p. then in Evans and Perkins [8] Theorem 5.1 it is shown that for all t > 0 |J S(n* k * Xt) C S(Xt) P m - a.s. t=i where //** is the k'th fold convolution of (i with itself.For a super symmetric stable process this implies S(Xt) = 0 or R d P m - a.s. Vr > 0. (3.78) Similar results for certain Feller processes are obtained . Consider a Markov jump process with bounded generator A so that Af(x) = p f M * , dy)(f(y) - f(x)) (3.79) JE with p > 0 and \x a probability kernel such that x —* f it(x,dy)f(y) G Co(E) for all / G Co(E). Then for t > 0 [J^S^J ...jXt(dx1)ti(xudx2)...Li(xk,-)j C S(Xt) P m - a.s. (3.80) We shall show that (3.78),(3.80) are far from being sample path properties and that near the time of death there will be exceptional times at which the support is concentrated arbitrarily close to the death point. We start by examining the case where the spatial motion is a Markov jump process as described above. Note that Af is well defined by (3.79) for any bounded measurable / . A monotone class argument shows that for any bounded measurable / the process Xt(f) is continuous and satisfies the usual semimartingale decomposition. 59 Theorem 3.5 For all £ > 0 , with probability one there exist distinct tn j £ such that S(XtjCB(F,e). PROOF . Take ACE and let / = l(x G A). We shall use the time changed process Yt(f) as in section 3.2. Let Bt be an independent Brownian motion defined if necessary on an extension of the original probability space. Define Bt= / t ( y , ( / ) ( i - y . ( / ) ) ) - 1 / 2 i(n(/)^o)d7v,(/)+ / ' i ( y , ( / ) = o)dS. Jo Jo so that Bt is a Brownian motion and Yt(f) = m(f) + f Y,(Af)ds + / ' ( y . ( / ) ( l - Y,{f))yl2dB,. Jo Jo I f Y t ( / ) = 0 or 1 then Yt is supported on Ac or A respectively. So we look for times at which Y t ( / ) ( 1 — y ( / ) ) becomes zero. Fix iV G N and define Zt(f) — y v + « ( / ) ( l - Yff+t(f)). By Ito's formula we have Zt(f) = Z0(f)+ f(l-2YN+,(f))(YN+s(f)(l-YN+,(f))y'2dBs Jo + ( (l-2YN+.(f))YN+,(Af)ds- f YN+,(f)(l-YN+t(f))ds Jo Jo = Zo(f) + A/3. - Z,{f))ds + [\z.(f)(l - *Zt{f)))l!2dB. Jo Jo where P. = (l-2YN+,(f))Y,(Af) Bt = / sgn(l - 2Y,(f))dB, JN so that Bt is another Brownian motion. Since the function — 4a:)) 1 / 2 satisfies the Yamada-Watanabe criterion (see Rogers and Williams [20] Theorem V.40.1 ) we have a unique solution on the same prob- ability space to the stochastic differential equation Xt = Z0(f) + f((l/8) - Xs)ds + f \X,{\ - AX,)\ll2dBs. Jo Jo Lemma 3.6 shows that Xt takes values in [0,1/4] and zero is a recurrent point . Define TN = inf(< > 0 : Y j v + t ( l ) > ( l / 8 p / | | ) ) which is a GN+t stopping time. For s <TN | A | = | ( l - 2 y w + i ( / ) ) y A f + . ( y l / ) | < l / 8 . 60 So by a comparison Theorem for one dimensional diffusions ( see Rogers and Williams [20] Theorem V.43.1 ) we have Zt{f) < Xt for t<TN Pm - a.s. (We have applied the comparison Theorem up to a stopping time .The changes needed in the proof of Theorem V.43.1 are easy. ) Since zero is recurrent for Xt , on the set {Tyv = oo}, Zt(f) must hit zero infinitely often as t —*• oo. Since Y,(l) —• 0 as s —• oo , P(T/v = oo) t 1 as N —• oo. So with probability one there exist tn T oo so that Ytn(f) = 0 or 1. (3.81) Given e > 0 let (Am)m be a countable collection of open balls of radius e/2 that cover E . Fix w so that (3.81) holds simultaneously for all fm = I(z 6 Am). Find mo(w) so that F(u) €'Amo(uy Since Yt —• 8f then Yt(Amo) —+ 1 . So there exist i „ f oo so that Yi n ( I(x 0 Amo)) = 0 and 5 ( X 0 ( n ) = S ( Y t J C B (F , e) for all n. • Lemma 3.6 Let Bt be a Brownian motion defined on a probability space (Q.,T,P). Let Xf be the unique solution to the stochastic differential equation dXt = ((l/8)-Xi)dt + \X,(l-4Xt)\1,2dBt (3.82) X0 = *€ [0,1/4] Then P(3n such that Xt > 0 for all t > n) = 0 . PROOF. Equation (3.82) is pathwise exact so we may find a pathwise unique solution on any space and any two solutions have the same law. Let Pz be the law of Xx on path space. Then the laws (Px)x form a strong Markov family. We write xt for the coordinate function on path space. Let Yt be the unique solution on (Q,^ 7 , P) to the S.D.E. dXt = ((l/S)-Xt)/\0dt-r\Xt(l-4Xi)\1/2dBt X0 = 0 By uniqueness Yt = 0, Wt > 0. So by a comparison Theorem (see Rogers and Williams [20] Theorem V.43.1 ) Xx > 0, V< Pm — a.s. We may treat the boundary x — 1/4 similarly and conclude P*(z, €[0,1/4], V r > 0 ) = l 61 It is enough to show that there exist <o s ° that P 1 / 4 (30 < t < t0, xt = 0) = c> 0 (3.83) for then by the strong Markov property Px(30 < t < to, xt = 0) > c for all x G [0,1/4] and setting An = (3t G (n<o,(n+ 1)<0], xt = 0) we have P x(An\Au... , ^ „ _ i ) > c and P x(An i.o.) = 1 which implies the result. On the interval [6, (1/4) — 6] where 0 < 6 < 1/8 will be chosen later , we can construct a weak solution to (3.82) using a scale and time change of Brownian motion in a standard manner (see Rogers and Williams V.44 .) We shall then examine the behaviour near the endpoints seperately. Set s(x) = f (u(l - 4u))- 1 / 4du for x G [6, (1/4) - 6] so that s(x) is a strictly increasing C2 function taking [5,(1/4) — 6] —• [a,b]. Set h(x) — s'(x)x(\ — Ax) and g(x) = /i(s - 1(x)) . Then g is a continuous function on [a, 6] bounded away from zero by a constant K. Let Bt be a Brownian motion on (fl, T, P) started at s(x) G [a, b] . Set Ty = inf(t > 0 : Bt = y) and At = f g(Bu)-2du for t < fa A f j . Jo Let jt be the continuous strictly increasing inverse to At- Then Yt = Blt solves the S.D.E. dYt = g(Yt)dBt for some Brownian motion Bt and Zt = s - 1 ( Y t ) is a weak solution to (3.82) up till the time inf(t > 0 : Zt G {&, (1/4) - 6 » . Now for t0 > 0 P( f a < f» < t0) > 0 so if Ty — inf (/ > 0 : xt = y) then P*(Ti < T ( 1 / 4 ) - « < #2<o) > 0 Vx G (5, (1/4) - 6). (3.84) For the behaviour near x = 1/4 we need only that we can find to,6 > 0 so that ^ 1 / 4 (x < 0 < (1/4) - 26) > 0 (3.85) and this follows since Xt = 1/4 is not a solution to (3.82). For the behaviour near x = 0 we use another comparison . Let Yt = (1 + cos(2B<))/8 where Bt is a Brownian motion started at xo and 6 = (1 + cos(2xo))/8. Ito's formula gives Yt = 6 + 2 f\(l/S)-Y,)ds+ /V«(1 -4Y. ) ) 1 / 2 <W, Jo Jo 62 where Wt is another Brownian motion. Let Xt be a solution on the same space to the equation (3.82) with x = 1/4 and Wt the Brownian motion. Then the comparison Theorem shows Xt < Yt up till the time T = inf(i >0:Xt/\Yt = 1/8) = inf(i > 0 : Yt = 1/8). But the construction of Yt implies that there exists t0 such that with positive probability , Yt = 0 for some i < to < T. So P\To < T1/8 < t0) > 0. (3.86) Equation (3.84),(3.85),(3.86) together imply (3.83). • Example. We examine the simplest nontrivial superprocess. Let E = {a, b] and the underlying spatial motion be a Markov chain leaving each state at rate one. Then if we write Xt(a),Xt(b) for Xt({a}),Xt({b}) the martingale problem reduces to a pair of linked stochastic differential equations. Xt(a) = X0(a)+ f\x,(b)-X,(a))ds+ f {X,{a)fl2dBa, (3.87) Jo Jo Xt(b) = X0(b)+ f\x,(a)-X,(b))ds+ l\x.(b)yl2dB\ (3.88) Jo Jo where Bf,Bf are independent Brownian motions. So we consider the superprocess as a diffusion on R 2 . Let D = ((x,0) : x > l/2)U((0,y) : y > 1/2) . We will show that with probability one (Xt(a),Xt(b)) never hits D.Define Dr = ((*,()):*> (1/2)+ r) Rr = ((x,y):y>x-(l/2)-r) It will be enough to show P((Xt(a),Xt(b)) 6 DT for some t > 0) = 0 for all r > 0. The properties of the a-dimensional Bessel process ( see Rogers and Williams [20] V.48 ) show that if Zt satisfies dZt = adi + (Zt)l,2dBt (3.89) then for a > 1/2 , P(Zt > 0,V< > 0) = 1 and a = 1/2 is critical. By comparing (3.88) to the S.D.E. (3.89) solved on the same space with respect to B\ we see that Xt(b) > 0 up till the time So = inf(t > 0 : {Xt(a),Xt(b)) G R0). Define Tn = i n f ( i > 5 „ _ i :(Xt(a),Xt(b))€R°/2) Sn = inf(< >Tn : (X t (a ) ,X t (6 ) )Gl 0 ) 63 Xt(b) 1/2 R Xt(a) 1/2 Figure 3.3: Typical sample path of (Xt{a),Xt(b)). Then by the strong Markov property and the same comparison argument Xt(b) > 0 for Tn < t < S„. By continuity of paths T„ | oo and the result is proved. The finite lifetime of the process implies that the diffusion converges to (0,0) and Theorem 35 shows that it approaches the origin in a particular manner . There exists R(ui) > 0 such that inside 2?((0,0), R) the diffusion will not hit one axis and will hit the other axis at an infinite number of points that accumulate at (0,0). We would like to extend Theorem 3.5 to superprocesses with more general spatial motion. We take one step in this direction by showing that the semimartingale decomposition for Xt(H) in section 3.1 allows us to extend Theorem 3.5 to super symmetric stable processes of index a < 1/2. Note also that the result is true for super Brownian motion (see Liu [15] where it is shown that the diameter of the support of super Brownian motion converges to zero at extinction ). L e m m a 3.7 Let Xt be a one dimensional super symmetric stable process of index a started at m 6 M F ( R ) . IfO<a,0< 1/2 then the patht —• Xt(\x\~p) is continuous on (0,oo). P R O O F . We use a stopping time argument similar to that in Perkins [16] Proposition 4.4. Take the nonstandard model with « m. Find a,/3 so that 0 < max(a,/3) < /? < a < 1/2. Write Bn for B(0,2~") and rB for (rx : x e B). Fix integer M > 1 and define Tn = inf(t e F f l [1/M,oo) : NL(Bn) > 2.2-""). 64 Let t? = j2~n*,/; = [t?, t? + 1 ) . Then P(T„e/;) = P (T„ ei?,NtUi(2Bn)> 2-"?) + P(Tn£l»,^ £ I ( ^ g 5 „ , ^ + i i i A ) < 2 ^ ) (3.90) + P(T„ £ z;,^-1 £ I(yV7n G B„, |7V7V - yV?„| > 2"n) > NTn{Bn)2- nP) Denote the terms on the right hand side of (3.90) as 1,11 and III.For t" > 1/m / < P(iV < ; + i ( 2 B n ) > 2 - ^ ) < 2 n'E(Xt7+i(2Bn)) < 2 m ( l ) M - 1 / o 2 - n ( 1 - ^ . T„ is a At stopping time so by the strong Markov property II = E[l(Tner?)-pNTjw\^ £ W ^ » , « 1 - T . M r A ) < W / 2 ) 7 ~ « ; + 1 -T.(«) < P(T„ € /;)exp(-2"( 5-^/4) using Perkins [16] Lemma 4.1.a. Similarly III < E \ l ( T n e I ? ) P " T " ( u \ n - 1 Y  l ( \ N k - T ^ ) - ^ \ > ^ n ) > N T n ( u ) ( B n ) - 2 ^ ) \ < P(T-neI?)2.P0(\Y2-n*\>2-n) < CP(Tn e /;?)2-"(*-a) using Perkins [16] Lemma 4.2.a. Summing (3.90) over 7" C [1/M, M] we have P(T„ <E [l/M,M]) < C(2-" ( 1 - 5 - ' 5 ) + exp(-2"(5-^) + 2- n ( i - a >) which sums over n. So for large n , for all t 6 [1/M, M] Xt((-2-n,2-n)) < 2.2~nK Thus Xt(\x\-P) is uniformly bounded for t G [1/M,M] and (X t(|a:| _ / 3 A n) : n = 1,2,...) is a Cauchy sequence in C [ l / M , M], Pm — a.s. So Xt(\x\~P) is continuous on [ l /M.M] for any M. • 65 Remark. In Perkins [16] Theorem 6.5 it shown that if a < 1 then there exist constants 0 < eg < c 9 < oo such that for any m G Mp(R) , setting 4>a{x) = x<Mog+log+ l/x At = (z : HmsupXt(S(a;,a))^0(a)_ 1 G [cg,c9]) ajo then Xt(Act) = 0 ,V i>0 ,Pm -a.s. We call a point in At a point of density for Xt- For a < 1 it follows from the facts that i. A« is Lebesgue null for all t > 0 (Perkins [16] ) ii. The laws of super symmetric stable processes on <r(X, : s > to > 0) are equivalent under translation of initial measures (Evans and Perkins [8]) that for any fixed x , P(x is a point of density at some t > 0) = 0 . This also follows ( for a < 1/2 ) from Lemma 3.7 for if 0 G A< then A"t(|x| - a) = oo. Contrast this with the fact that equation (3.78) implies that for a fixed t > 0 the points of density are dense in R. Proposition 3.8 If Xt is a super symmetric stable process of index a < 1/2 tn dimension one started at a finite measure then the conclusions of Theorem 3.5 still hold. PROOF. Fix an open ball B = (a, 6) of finite radius in R. From Theorem 3.2 and the following remark we have the decomposition Xt{B) = X0(B) + Vt + Mt(B) where (M(B))t = /' X,(B)ds and Vt has finite variation on [S,T] for any 0 < S < T < oo. Define { limh_o+(Vr<+jl — Vt)/h if this limit exists 0 otherwise We will find an upper bound on Note that for any 6 > 0 , T(IB(X) is a C°° function vanishing at infinity. Let g(x) = sup < > 0 |AT<Is(x)|. Scaling arguments show there exists C such that g(x) < C(\x - a\~a + \x - b\-a). For fixed 0 < s < t \Xt(TsIB) - X,(T6lB) - Mt{T6lB) + M,(T6lB)\ = | j Xr(ATtlB)dr\ < J Xr(g)dr. 66 Letting 6 | 0 \Vt -V,\ = \Xt(B) - X,(B) - Mt(B) + M,(B)\ < J Xr(g)dr. Xr(g) is continuous and bounded by Lemma 3.7 so Vt is absolutely continuous and \vr \ < Xr(g) for a.a.r Q n a.s. Now we follow the proof of Theorem 3.5. Recall that t= I l/Xr(l)dr (3.91) Jo and we set Yt(B) = XD,(B)/XDt(l) and Zt{B) = Yt(B)(l - Yt(B)). Then Zt(B) satisfies Zt = Z0+ [\(l-2YN+s(B))vDN+,-Zs)ds+ j\zs{B){\-AZs(B))yi2dBs. Jo Jo Now the comparison argument of Theorem 3.5 will work provided we can show P(\VDN+.I < 1/8 for a.a.s) — 1 as yV oo (3.92) But from (3.91) we have \t-s\ < \Dt-D,\max(Dt<r<Dt)(l/Xr(l)) so that |vn r | < XDr(g) for a.a.r. Qm- a.s. From Lemma 3.7 XDr(g) —*• 0 as r —• oo and (3.92) follows. • 3.4 Recovering the spatial motion How much can you tell about the underlying spatial motion from a single path of a superprocess? In the following result we use an arbitrarily short piece of the path but recover only partial information. i Lemma 3.9 Let X% be a superprocess started at m E Mp(E) with spatial motion a Feller process with generator A . For f € D(A) satisfying m(|/|) = 0 and Var{Xt{f)) = 0(t2) as t - 0 (3.93) there is a sequence tj [ 0 such that n {l/n)Y,tJ1XtiV)*™W)- (3-94) P R O O F . Set Zt = (l/t)X t(/).Then E(Zt) = (l/t)(m(/)+ f E{X,{Af))ds Jo = (1/0/ mT.{Af)da-> m{Af) as < 0 (3.95) Jo 67 Hypothesis (3.93) ensures that Var(Z<) remains bounded so it remains only to show that we can pick tj I 0 fast enough that the Z < y ' s are nearly uncorrelated. Using a product moment formula (Dynkin [6] Theorem 1.1 ) we have for s < t Cov(Z„Zt) = (l/st) f dm f Tr{Tt-rU)T-rU))dr JE JO < (l/0||/||mT.(|/|). Note that mTa(|/|) —• m( | / | ) = 0 as s —• 0. Now pick to > 0 arbitrarily small and t„ inductively so that Cov(Ztn,ZtJ<2-\m-»l (3.96) Then £ ( ( ( l / n ) ] [ > . - m ( A / ) ) 2 ) i = i = (1 /n 2 ) JT E((Zt. - m(Af)f) + (1 /n 2 ) £ Cov(Zt,, Ztj) n +(l/n2)YE(Zti - m(Af))E(Ztj - m(Af)) (3.93),(3.95),(3.96) ensure that all three terms go to zero. • Remark. Since , (l/t2)Var(Xt{f)) = (1 / t 2 ) / ' mTr(T2_r(f))dr < (l/t)mTt(f2) Jo a sufficient condition for hypothesis (3.93) to hold is that / 2 € D(A) for then ( l / t ) m T t ( / 2 ) = (1/t) f mTr(A(f2))dr < p ( / 2 ) | | m ( l ) . Jo I f this condition holds then we may take tn = 2~" in (3.94). As an example we take the underlying motion to be a pure jump Levy process on the line (Yt : t > 0). Hence £(exp(-»0Y«)) = exp(t / {eiBx - 1 - (i0x / ( l + x2)))n{dx)) JTl where fi, the Levy measure , gives finite mass to (—a,a) c for any a > 0. We show that from any initial segment of a path of the super Levy process Xt started at 6o we can recover the Levy measure /z. Fix a > 0 and let / = I(a,oo) Then although / 0 D(A) i t can be shown that (T, / (0) - / ( 0 ) ) / t = P(Yt € (a, oo))/t - / i(a, oo) as t - 0. (3.97) ' 68 r Find t0 so that P(\Y,\ > a/2) < 2sfi([-a/2,a/2]c) for all s < t0- Var(Xt(f)) = fTT(T2_rf)(0)dr Jo < [ P(\Yr\>a/2) + (P(Yt-r<E(a/2,oo))2dr Jo < 2fi([-a/2> a/2]e) j (r + (t - r)2)dr = 0(t2) (3.98) Jo The proof of Lemma 3.9 shows that the bounds (3.97),(3.98) together will imply n Sn := ( l / n ) £ V ' X 2 - y ( / ) ^ ,1(0,00) and along a subsequence (njt)* therefore Snk —* /i(a, oo) almost surely . From a countable number of intervals all bounded away from zero we can recover the entire measure fx in this way. Corollary 3.10 Suppose Ai,A2 are generators of two conservative Feller processes on E and that there exists f e C(E) satisfying f,f2 £ D(AX) D D(A2) and Aif(x) ^ A2f(x) for some x £ E . Let Pi be the law of the superprocess with spatial motion generated by Ai and started at 8X .Then P\ and P2 are singular. P R O O F . For conservative Feller processes we have Ail = 0 so that replacing / by / — f(x) £ D(Ai) we may assume that f(x) = 0 . The proof of Lemma 3.9 and the following remark shows that we can find an explicit subsequence (nk)k such that Pi lives on the measurable subset Q x = { « : ( l / n t ) E 2 ' w 2 - ' ( / ) Aif(x) as k co) Since (l/n*) 2 jW2-J(/) will have a subsequence that converges almost surely to A2f(x) , under P 2 we have P2(fti) = O.D We return to the example of section 3.3 to show that it may not be possible to recover the entire underlying motion. Let E = {a,b},Q = D([0,oo),MF(E)),Xt(u) = u{t),T = <T{X, : s > 0). Let P ^ 0 ^ be the law on CI of the superprocess with spatial motion a Markov chain on E with generator Af(a) = —Af{b) = rf(b) — f(a) and started at ao6a + 606j .Thus the 'particles' jump from a to b at rate one and from b to a at rate r. Proposition 3.11 For r i , r 2 > 0 there exist ao,6o > 0 such that p^0,6") and Pr°°'b°^ are not singular measures. 69 PROOF . We write Xt(a),Xt(b) for Xt({a}),Xt({b}). Define {inf(0 < t < £ : Xt{a)/Xt(l) < 1/2) if this set is non-empty +00 otherwise We will find M > 0 so that if ft0 = {T = + o o , £ < M ) 8 up,< e X,(l) < M] then P ^ 0 ' 4 ^ ^ ) > 0 and then give an explicit Radon-Nikodym derivative for From the characterisation of the law of the death point ( equation (3.76)) we have Pr]'*\F = a) > 0. So P£'lHxt(a)/Xt(l)-+l)>0. Pick n so that if = inf(t > 0 : Xt(l) < 1/n) then P^HXtW/Xtil) > 1/2 for all < t < 0 > 0. Then by the strong markov property if F(x,y) is the distribution function of the pair (Xin(a),Xin(b)) under Pr]'1^ then 0 < r r P^V\T = +cc)aT:r, y. Jo Jo So we may pick a0,b0 so that p r ( | ° 0 ' l o ) (T ' = +00) > 0 and since { < 00 and sup, < { Xt(l) < 00 almost surely we may fiad M so that P r ( I 0 o ,* o )(fi 0) > 0. Let R = T A inf(t > 0 : X,(l) > Af) A inf(t > M : X,(l) > 0) where we let inf(0) = 00. Under Pi ( o 0 , * o ) J rtAR ' ( r x X . ( 6 ) - X . ( o ) ) d « 0 M*(b) = X 4 A « ( 6 ) - 6 0 - / (^ . (oJ-rxX.t*))^ Jo are martingales satisfying <A/*(a))« = ft** X.(a)ds, (XR(b))t = / 0 ' A f i X,(b)ds and (MR(a), MR(b))i = 0. Define = /V2 - r 1 )X.( 0 ) /X,( a )dM, f i (a) + '/Vi " r2)dM,R(b) Jo Jo V t = 5(Z t) = exp(Z« - (1/2)<Z),) Then »tA.R (Z), = / (r, - r 2) 2((X. 2(6)/X.(a)) + X.(6))ds. Jo 70 For t < T , X,(b)/X,(a) < 1 so that (Z)t < M 2 ( r 2 - r^) 2 for all t . This ensures that % is a uniformly integrable martingale ( see Elliot [7] Theorem 13.27 ) . Define Q by dQ/dP^a'ha) = n T O Then (see Rogers and Williams [20] Theorem IV.38.4 ) under Q ft A H t R(a)-{MR(a),Z)t = XtAR(a)-°-o- / (r2X,(b) - X,(a))ds Jo J ftAR ' (X,(a)-r2X,(b))ds o are martingales with the same brackets processes as MR(a), MR(b). This characterises the law of Q on TR which must agree with P ^ 0 , B O ) . But Q0 - { R = oo} so if A C ft0 then A = An { R = oo} e FR • Thus Q\no = P r (r ' b o )|n„.D 71 B i b l i o g r a p h y [1] Albevario.S.,Fenstad.J.E.,Hoegh-Krohn.R. and Lindstrom.T. (1986) Nonstandard Methods in Stochastic Analysis and Mathematical Physics, Academic Press. [2] Anderson.R.M. and Rashid.S. (1978) A nonstandard characterization of weak convergence, Proc.Amer.Math.Soc. 69, 327-332. [3] Cutland.N.(1983) Nonstandard measure theory and its applications, Bull. London Math. Soc. 15, 529-589. [4] Dawson.D.A. and Hochberg.K.J. (1979) The carrying dimension of a stochastic measure diffusion, Ann. Probab. 7, 693-703. [5] Dawson.D.A., Iscoe.I. and Perkins.E. (1988). Super-Brownian motion: path properties and hitting probabilities. To appear in Prob.Th.Rel.Fields. [6] Dynkin.E.B. (1988) Representation for functionals of superprocesses by multiple stochastic integrals, with applications to self-intersection local times.To appear in Asterisque. [7] Elliot.R.J. (1982) Stochastic Calculus and Applications , Springer,Berlin. [8] Evans.S.N. and Perkins.E. (1989) Absolute continuity results for superprocesses with some applica- tions.Preprint. [9] Falconer.K.J. (1985) The Geometry of Fractal Sets, Cambridge tracts in mathematics 85, Cambridge University press. [10] Fitzsimmons.P.J. (1988) Construction and regularity of measure valued branching processes.Israel J.Math. [11] Harris.T.E. (1963) The Theory of Branching Processes, Springer-Verlag, New York. [12] Kauffman.R. (1969) Une propriete metrique du mouvement brownien. C.R.Acad.Sci.Paris. 268, 727-728. 72 [13] Kingman.J.F.C. (1973) An intrinsic description of local time. J.London.Math.Soc.fS.StT.2,725-731. [14] Konno.N. and Shiga.T. (1988) Stochastic differential equations for some measure valued diffusions. To appear in Prob. Th.Rel. Fields. [15] Liu.L. (1988) Processes before extinction and comparison of measures by counting atoms. Ph.D. Thesis . University of Rochester. [16] Perkins.E. (1988). A space-time property of a class of measure-valued branching diffusions. To appear in Trans. Amer. Math. Soc. [17] Perkins.E. (1988) Polar sets and multiple points for Super-Brownian motion. To appear in Ann.Prob. [18] Perkins.E. (1988). The Hausdorff measure of the closed support of super Brownian motion.To appear in Annates de I'Instilute Henri Poincare. [19] Roelly-Coppoletta.S. (1986) A criterion of convergence of measure-valued processes: application to measure branching processes, Stochastics 17,43-65. [20] Rogers.L.C.G. and Williams. D. (1987) Diffusions,Markov Processes , and Martingales ,Volume 2: ltd Calculus. Wiley. [21] Sawyer.S. Branching diffusion processes in population genetics. Preprint. [22] Shiga.T. (1988) A stochastic equation based on a Poisson system for a class of measure valued diffusion processes. Preprint. [23] Sugitani.S. (1988) Some properties for the measure valued branching diffusion process. Preprint. [24] Walsh.J.B. (1986) An introduction to stochastic partial differential equations.Lecture notes in Math- ematicsll80 (Springer Verlag) 265-439. [25] Watanabe.S. (1968) A limit theorem of branching processes and continuous state branching pro- cesses. J.Math.Kyoto. Univ.%, 141-169. 73

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