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Interacting measure-valued diffusions and their long-term behavior Gill, Hardeep Singh 2011

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Interacting Measure-Valued Diffusions and their Long-Term Behavior by Hardeep Gill B.Sc. Pure Mathematics and Statistics, Calgary, 2005 M.Sc. Mathematics, University of British Columbia, 2007 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF Doctor of Philosophy in THE FACULTY OF GRADUATE STUDIES (Mathematics) The University Of British Columbia (Vancouver) August 2011 c© Hardeep Gill, 2011 Abstract The focus of this dissertation is a class of random processes known as interacting measure-valued stochastic processes. These processes are related to another class of stochastic processes known as superprocesses. Both superprocesses and inter- acting measure-valued stochastic processes arise naturally from branching particle systems as scaling limits. A branching particle system is a collection of particles that propagate randomly through space, and that upon death give birth to a random number of particles (children). Therefore when the populations of the particle sys- tem and branching rate are large one can often use a superprocess to approximate it and carry out calculations that would be very difficult otherwise. There are many branching particle systems which do not satisfy the strong in- dependence assumptions underlying superprocesses and thus are more difficult to study mathematically. This dissertation attempts to address two measure-valued processes with different types of dependencies (interactions) that the associated particles exhibit. In both cases, the method used to carry out this work is called Perkins’ historical stochastic calculus, and has never before been used to investi- gate interacting measure-valued processes of these types. That is, we construct the measure-valued stochastic process associated with an interacting branching parti- cle system directly without taking a scaling limit. The first type of interaction we consider is when all particles share a common chaotic drift from being immersed in the same medium, as well as having other types of individual interactions. The second interaction involves particles that at- tract to or repel from the center of mass of the entire population. For measure- valued processes with this latter interaction, we study the long-term behavior of the process and show that it displays some types of equilibria. ii Preface The work provided below was conducted solely by myself, under the supervision of my advisor Professor Edwin A. Perkins. The problems investigated were orig- inally conceived by Professor Perkins. I proved the results (solutions of those problems), wrote the research articles and subsequently submitted them to journals of my choosing. This dissertation is based on two publications that resulted from my time as a doctoral student: (1) Gill, H.S. A super Ornstein-Uhlenbeck process interacting with its center of mass. To appear. (2) Gill, H.S. Superprocesses with spatial interactions in a random medium. Stochastic processes and their applications. December, 2009. 119 (12): 3981-4003. Publication (1) forms the basis of Chapter 3, and (2) forms the basis for Chapter 2. iii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Branching particle systems and superprocesses . . . . . . . . . . 2 1.2 Superprocesses in a random medium . . . . . . . . . . . . . . . . 5 1.3 An interacting super Ornstein-Uhlenbeck process . . . . . . . . . 8 2 Superprocesses with spatial interactions in a random media . . . . 12 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3 The strong equation . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.4 A martingale problem . . . . . . . . . . . . . . . . . . . . . . . . 35 2.5 Additional properties of solutions . . . . . . . . . . . . . . . . . 40 3 A super Ornstein-Uhlenbeck process interacting with its center of mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.3 Existence and preliminary results . . . . . . . . . . . . . . . . . . 50 iv 3.4 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.5 The repelling case . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 v Acknowledgments There were many people who made this dissertation possible. Some of them im- pacted it directly, and some of them kept me in “Chahrdi Kala,” which in Punjabi means in high spirits. My advisor, Professor Edwin A. Perkins, has my deepest gratitude for his pa- tient and thoughtful guidance. I am especially grateful for the research problems he suggested, which provided me with a great deal of enjoyment and an invaluable source for learning. I quickly found that Ed is a veritable fount of ideas for those difficult times I was stuck while working on a problem. I also appreciate his careful readings of this manuscript and the papers which resulted from it. I would like to thank the members of my supervisory committee, particularly Martin Barlow and John Walsh, for their insightful comments and suggestions. I am grateful to Jie Xiong for suggesting additional avenues of research. My career path might have been completely different without Ms. Williams, my high school math teacher. Her creative and challenging class fostered my inter- est in mathematics. Kiran has my gratitude for her encouragement and support during the writing of this manuscript. This was nowhere more evident than by her willingness to undergo an allergic reaction rather than leave my doctoral defense room. Kiran, Lisa, Sidd, and my brothers Gurbeer and Harbir have my thanks for interspersing my studies with pleasant diversions which helped to keep me going. Lastly, I would like to thank the many other people with whom I have interacted during the course of this work, for making my stay at UBC vibrant and enjoyable. vi Dedication Dedicated to my parents, for their unconditional love and support. Their perseverance and hard work allowed me the opportunity to pursue my aspirations. vii Chapter 1 Introduction Branching processes are a well studied branch of probability and have been of in- terest since Galton and Watson first considered the problem of extinction in family names, in the late part of the nineteenth century. The process that Galton and Wat- son first studied consisted of particles that die after each generation leaving behind an independent and identically distributed number of offspring. If one considers a large Galton-Watson population, then its size can be approximated by what is known as Feller’s continuous state branching process. This in turn makes it much simpler to determine various properties of the original Galton-Watson process. The goal in what follows is to construct and study processes that act as analogues of Feller’s continuous state branching process for more complicated branching parti- cle systems which have interactions between particles. A branching particle system (BPS) can essentially be thought of as a Galton- Watson branching process where each particle alive at a given time is also assigned a randomly changing position in space. There are many examples of such systems in biology: consider a single celled bacteria which has the ability to move using its flagellum and which reproduces through binary fission. If each individual bac- terium is not affected by the location of the other bacteria, this could be considered a non-interacting BPS. A second example is a larger species called phytoplankton which live in the ocean. Each plankton moves on some random path determined by its internal motor and reproduces (branches) randomly. The plankton spatial posi- tion is affected both by individual interactions (e.g. reproductive) with each other 1 as well as by a common drift due to the water current and therefore this system is an example of an interacting BPS. It can be computationally difficult to simulate a BPS when the population is large since this involves simulating the random paths for each of the particles. Fortunately, if for a BPS the number of particles is large and we observe the sys- tem over a large period of time, then it can be approximated by a measure-valued stochastic process. Equivalently, the behavior of a large population, rapidly branch- ing, BPS can be inferred from the properties of the corresponding measure-valued process. A mathematically rigorous result of this type was established indepen- dently by Watanabe [25] and later by Dawson [1] for non-interacting BPS, where the associated measure-valued process is a superprocess. The situation for inter- acting BPS is much more difficult to ascertain since different interactions require different methods of examination. Hence the results in this direction tend to be proven on a case-by-case basis. Chapters 2 and 3 study branching particle systems with two different types of interactions. We construct the scaling limits of these systems explicitly using something that is known as “Perkins historical stochastic calculus,” which is a the- ory pioneered in the early nineties by Edwin Perkins [13] to study many types of interacting measure-valued processes. 1.1 Branching particle systems and superprocesses Before mathematically describing an interacting particle system, we define a non- interacting branching particle system. Suppose the spatial motion of a typical par- ticle follows the stochastic process {Yt : t ≥ 0}, which takes values in a space E (usually E is Rd). Also suppose that when the particle dies it branches into k par- ticles with probability ν(k). Let Λ be the set of all particles in the particle system. We assume that a particle, labeled by α , born at a (possibly random) time τ(α), lives until time d(α) whereupon it dies after giving birth to Mα offspring, where {Mα : α ∈ Λ} are independent and identically distributed random variables each with law ν . Typically, the lifetime of a particle, `(α)≡ d(α)− τ(α) is taken to be an independent exponential random variable, or to be constant. 2 We give ourselves the family of stochastic processes {Yα(t),τ(α)≤ t < d(α) : α ∈ Λ} which will describe the paths of the particles. In particular, for particle α we will insist that Yα(τ(α)) =Yα ′(d(α ′)−), where α ′ is the parent of α . This simply says that where the parent α ′ died is where the particle α was born. We will also assume that the particles alive at a given time t move independently of each other. Then one can define a process Z which counts the number of particles in a region of space at a given time: For A a subset of E, define for t > 0, Zt(A) = ∑ α∼t δYα (t)∈A, where α ∼ t denotes that particle α is alive at time t. Z0 is defined by the particles alive at time 0, which we will take to be given. Assume in the above that Y is a nice process (say a continuous Markov pro- cess). Suppose the branching law νN has mean 1+β/N and variance γ > 0, where β and γ are fixed numbers. Assume that each labeled particle α is born at an in- tegral multiple of N−1 and lives for a time `(α) = N−1. Suppose also that we are given a finite measure m(·) on E. Then for A⊂ E and t > 0, let XNt (A) = 1 N ∑α∼t δYαt ∈A, (1.1) where XN0 (·) is given by a Poisson point process with intensity Nm(·). This gives rise to the aforementioned result: Proposition 1.1.1 (Watanabe, 1968; Dawson, 1975). There is a measure-valued, Markov process X such that XNt −→ Xt , for all t, in distribution with X0 = m. The process X is called a (Y,β ,γ)-Dawson-Watanabe superprocess. This prop- osition says that to estimate the probability of most types of events for a large population BPS, one can simply calculate the probability of such events for the corresponding superprocess (which is typically easier). The value of β determines whether the process is subcritical or supercritical. If β > 0 then it is supercritical and there is a positive probability that the process 3 will exhibit exponential mass growth and survive indefinitely. If β ≤ 0, the process will die out in finite time almost surely. The case β = 0 is called critical and β < 0 is subcritical. One immediate property of X is that it is not an atomic measure, as the XN were. Indeed, if Y is one dimensional Brownian motion, Xt is almost surely absolutely continuous with Lebesgue measure. In higher dimensions, the support of Xt tends to be fractal-like. For these results and much more, Perkins [14] is a wonderful reference. The superprocess Xt is deficient in the sense that it only keeps track of the mass alive at time t and ignores any other genealogical information present in the associated branching particle system. To remedy this, we can keep track of not just the position of a particle at time t, but instead the entire path of the particle and those of its ancestors until time t: for B⊂C([0,∞),E) measurable, define HNt (B) = 1 N ∑α∼t δYα·∧t∈B. Here, for t < τ(α), we let Yαt be the location of the unique ancestor of α alive at time t. Note: we say y·∧t is the path y until time t and fixed thereafter. Then one can adapt Proposition 1.1.1 for a special case to show that HNt −→ Ht for all t, in distribution. We call H the (Y,β ,γ)-historical process. It is then easy to project Ht down at time t to recover the process Xt (where X is the (Y,β ,γ)-superprocess): For A⊂ E, Xt(A) = ∫ 1(yt ∈ A)Ht(dy). One of the special features of superprocesses that results from the indepen- dence of particles alive at a given time is a type of log-laplace equation. This is an equation that can be used to infer properties of the superprocess (and therefore the associated BPS) through the solution of a particular non-linear partial differential equation. More precisely, for certain positive functions φ : E → R, one can show that E ( e− ∫ φdXt ) = e− ∫ VtφdX0 (1.2) 4 where Vs ≡Vsφ is the solution to ∂Vs ∂ s = AVs+βVs− γ2V 2 s , (1.3) with initial condition V0 = φ , where A is a second order differential operator called the infinitesimal generator of the process Y . This connection allows one to calculate probabilities and infer properties of X from the behavior of V , and conversely use properties of X to establish results for the non-linear PDE (1.3) (see for example, Le Gall [11]). 1.2 Superprocesses in a random medium In the first chapter below, which was published in [8], we construct a family of interacting measure-valued diffusions that live in a random medium by solving a stochastic equation driven by historical Brownian motion. This work is intended to provide oceanographers with a probabilistic model for phytoplankton populations during an algal bloom (colloquially known as a “red tide”). During a bloom, the populations of plankton in a region increase rapidly to the point where one can actually observe them in the water by its reddish tinge. This can have negative consequences higher in the food chain as some of these plankton, such as dinoflag- ellates, release neurotoxins in the water which can then become concentrated in filter-feeding marine animals. When these animals are consumed by humans it can cause possibly serious illnesses like paralytic shellfish poisoning. As such, it is vital to predict the likelihood of algal blooms taking place at a given location. Plankton are by definition ocean drifters and range in size from less than a micrometer in width to possibly bigger than a centimeter. The position of a typical organism is affected by the current of the ocean on a large scale. On a smaller scale the position of a plankton is determined by its internal motor and by eddies and ripples of water. As the size of the plankton is typically very tiny, we may think of it as Brown originally thought of the motion of a pollen grain in a glass of water - that it is constantly being bombarded by millions of water molecules which affect its position in a random manner. Thus the motion of a small plankton may be modeled by a diffusion. 5 The first probabilistic work that could be used to model large plankton popu- lations was found in the Ph.D. dissertation of Wang, published in [23] and [24], though it was not motivated directly by such applications. Dawson, Li and Wang in [2] carry this work further to include a variable branching mechanism. The con- nection to plankton dynamics was made in the work of Skoulakis and Adler [19]. Young in his survey paper [27] for the oceanographic community, discusses similar models under the guise ‘Brownian bugs.’ Young et. al. in [28] provide more support for the models such as the one above by showing that the observed ‘patchy’ dis- tribution of organisms in the water can be explained by the effect of the branching (reproduction) rather than other physical or chemical cues. The technique used in each of [23], [24], [2] and [19] consists of defining a sequence of interacting branching particle systems and taking a high density limit, as in Proposition 1.1.1. Under this method the particle α follows a path Yαt on its life time, where Yα solves the following stochastic differential equation: Yαt = Y α τ + ∫ t τ σ1(Yαs )dB α s + ∫ t τ b(Yαs )ds+ ∫ t τ ∫ σ2(Yαs ,ξ )dW (s,ξ ) (1.4) where σi and b are coefficients satisfying some reasonable conditions and τ ≡ τ(α). Here the {Bα}α∈Λ are an independent family of Brownian motions and W is a space-time white noise term independent of that family. The white noise term is shared across all the particles and therefore acts as a random medium which affects all particles simultaneously (i.e. as the ocean does for plankton) . This also means that the spatial motion of the different particles will depend on each other since they all have a common component driven by the same white noise. The white noise W is an object arrived at by taking a scaling limit of a space- time lattice of random variables. We divide up space-time in to cubes of side- length N−1 and to each of these, we assign an appropriately scaled independent random (spatial) direction. Then for a set A in Euclidean space, W Nt (A) denotes the sum of all the random directions of all the space-time cubes located inside the set [0, t]×A. We then send N to infinity to get the random direction Wt(A). Hence one can think of white noise as a random field generated by the movement of a very large number of infinitesimally small particles moving about according to Brownian motion (such as what happens in a liquid). 6 Skoulakis and Adler in [19] assume the white noise W in (1.4) is really a Brow- nian motion (which is a very specific white noise) whereas Wang and his coauthors all assume that the drift term b = 0. Also note that even in the full setting of (1.4), the only interactions between particles are as a result of the common drift due to the random medium. That is, one would expect that for a plankton population there should be some interactions of the plankton with each other, which are not present here. More is known for the setting where W is a Brownian motion in (1.4) than in the generalized case since it is easier to work with a Brownian motion in place of a white noise. In fact Xiong in [26], by utilizing random duals, shows the existence of a log-laplace equation similar to that of (1.2) where the Vtφ in that equation is replaced by a solution of a non-linear stochastic partial differential equation similar to Equation 1.3. This allows one to perform calculations for the limiting measure-valued process in a manner similar to that used for superprocesses, which is remarkable since as mentioned earlier, superprocesses arise as limits of non- interacting BPS’s, whereas the BPS presented here has interactions. The approach employed in Chapter 2 takes an existing historical process K and an independent white noise W and uses them to construct another measure- valued diffusion much in the same way one takes Brownian motion and constructs diffusions by solving stochastic differential equations. In particular, we solve the following stochastic equation: Let K be a critical historical Brownian motion (the process constructed above with Y a Brownian motion and β = 0), and A measur- able. Then define (SE) (a) Zt = Z0+ ∫ t 0 σ1(Xs,Zs)dy(s)+ ∫ t 0 b(Xs,Zs)ds+ ∫ t 0 ∫ Rm σ2(Xs,Zs,ξ )dW (s,ξ ) (b) Xt(A) = ∫ 1(Zt(y) ∈ A)Kt(dy) One can think of (SE) (a) as a stochastic differential equation describing the mo- tion of a given particle driven by a certain path y and a white noise, and as the natural analogue to (1.4). One major difference is that (SE)(a) includes interac- tions between individual particles (for example reproductive interactions between plankton) whereas the models considering (1.4) do not. (SE)(b) gives a measure- 7 valued process by integrating over all paths y with respect to K. Since Kt typically puts mass on those paths y that resemble Brownian sample-paths, (SE)(a) is akin to an SDE where y is replaced by a Brownian motion. Note that the white noise W does not depend on y. The main theorem in this work shows that there exists a unique (strong) solu- tion to (SE): Theorem 1.2.1. Under some Lipschitz conditions for the coefficients σi and b, there exists a pathwise unique solution (X ,Z) to (SE). With this theorem in hand, additional properties of X such as the compact sup- port property (which shows that a particle cannot travel too far in finite time), strong Markov property and stability of the solutions are established. One can also show without too much difficulty that a sequence of interacting branching particle sys- tems determined by the formula (1.1), with spatial position for particle α given by (1.4), converge in distribution to the solution X of (SE). In the case where there is no spatial dependence of the coefficients, we recover the results of Wang, Skoulakis and Adler. 1.3 An interacting super Ornstein-Uhlenbeck process The second chapter of the thesis (forming the basis for the article [9]) is devoted to the construction and examination of a particular class of interacting measure- valued processes with a singular drift called the super Ornstein-Uhlenbeck (SOU) process interacting with its center of mass (COM). What this means is that the BPS associated with this measure-valued process has particles whose spatial motion is described by a modified Ornstein-Uhlenbeck process. The Ornstein-Uhlenbeck process is one of the simplest stochastic processes – originally introduced in 1930 by Ornstein and Uhlenbeck in [21] to model the velocity V of a particle in a medium with a certain friction coefficient. The process satisfies dV =−γV ds+dBs, where B is a Brownian motion and γ > 0. The thought was that since Brownian motion is not differentiable, one cannot talk about the velocity of a Brownian parti- 8 cle and hence it is better to model the velocity by a diffusion process directly. Two of the key properties are that this process is mean-reverting and has a stationary distribution that is a Gaussian random variable. The interacting particle system corresponding to this process is made up of particles that move in a Brownian fashion, but are attracted to or repelled from the COM of the system. That is, for a particle α in the corresponding particle system, its motion during its lifetime is described by Yαt = Y α τ(α)+ γ ∫ t τ(α) (Ȳ (s)−Yαs )ds+ ∫ t τ(α) dBαi (s), (1.5) where Ȳt = 1 #{α ∼ t} ∑α∼t Yαt is the center of mass and {Bα : α ∈ Λ} is an independent family of Brownian mo- tions. This constructs a branching modified-OU system Z as follows: Zt(A) = ∑ α∼t δYαt ∈A. (1.6) The particles in the system Z exhibit attraction to its COM when γ > 0 and repul- sion when γ < 0. The motivation for the model came from a paper of Engländer [3] where he constructs a branching particle system Z′ where each particle moves according to (1.5). The system starts with a single particle, and all particles live for one unit of time and die at the integral times leaving behind exactly two offspring particles. It is clear that the particle system will have exactly 2btc particles alive at time t. Engländer then proves various theorems about the long term behavior of the parti- cle system and shows that it achieves a natural equilibrium. He proceeds by first showing that the center of mass Ȳt converges as t approaches infinity and then that for φ : Rd → R nice, 2−n ∫ φ(x)Z′n(dx)−→ ∫ φ(x)PȲ∞(dx), (1.7) as n goes to infinity, almost surely, where PȲ∞ is a Gaussian random variable cen- 9 tered at the limiting value of the COM, Ȳ∞. This particle system is particularly simple in that one does not have to worry about the population of particles going extinct at a given time and the singularity that results in the definition of the center of mass, as is the case for the more general BPS, Zt . Nevertheless, one can make an analogy between this system and a super- critical BPS on its survival set, as in both situations the systems exhibit exponential population growth. Therefore one expects that on the survival set a general super- critical BPS should exhibit similar long-term behavior as in this binary branching case. Given this insight, it is not surprising that if one can construct the scaling limit of such a sequence of BPS’s then it should exhibit some form of equilibrium behavior. To make this mathematically sound we define (similar to (SE) above) the fol- lowing stochastic equation: Let K be a supercritical historical Brownian motion, (i.e. β > 0 in the definition). Then define the process (X ,Y ) by the solution of the following equation: (SE)∗(a) dYs = γ(Ȳs−Ys)ds+dys (b) Xt(A) = ∫ 1(Yt(y) ∈ A)Kt(dy) where A⊂ Rd and the COM is defined as Ȳs = ∫ xXs(dx)∫ 1Xs(dx) . We proceed to show that Ȳt converges almost surely and use this fact to prove the main theorem: Theorem 1.3.1. If Px∞ is the stationary distribution of an Ornstein-Uhlenbeck pro- cess with attraction to x at rate γ > 0, then on the survival set Xt(·) Xt(1) a.s.−→ PȲ∞(·), as t→ ∞. The convergence obtained here is in fact even stronger than the one Engländer 10 showed for (1.7) since here we have convergence in measure. The methods used to get this result differ considerably from those of Engländer due to the generalized setting being considered. The novelty of this work is that it is one of few cases where we can explic- itly determine the long-term equilibrium behavior of an interacting measure-valued process. Unlike previous applications of the historical calculus, we must use it here to prove uniqueness in a non-Lipschitz setting. 11 Chapter 2 Superprocesses with spatial interactions in a random media 2.1 Introduction We will consider the following stochastic equation: (SE) (a) Zt = Z0+ ∫ t 0 σ1(Xs,Zs)dy(s) + ∫ t 0 ∫ Rm σ2(Xs,Zs,ξ )dW (s,ξ )+ ∫ t 0 b(Xs,Zs)ds (b) Xt(A) = ∫ 1(Zt(y) ∈ A)Kt(dy) ∀A ∈B(Rd). The coefficients are such that σ1 : MF(Rd)×Rd → Rd×d , σ2 : MF(Rd)×Rd × Rm→ Rd×d and b : MF(Rd)×Rd → Rd , where MF(Rd) denotes the space of all non-negative finite measures on Rd . K is a historical super-Brownian motion on some space, y is a path in Rd , and W is a d-dimensional white noise (on R+×Rm) that is independent of K. The precise definition of a historical process can be found in [14] and the exact meaning of a stochastic integral with respect to y is given below in Section 2.2. Later we impose some Lipschitz-like conditions on b,σ1 and σ2 to use a Picard iteration argument. Note that Z takes values in the space of paths in Rd , while X takes values in the space of paths in MF(Rd). 12 One can think of (SE) (a) as a stochastic differential equation describing the motion of a given particle driven by a certain path y and a white noise, and (SE)(b) as the measure-valued process obtained by integrating over all paths y with respect to a historical super-Brownian motion. As mentioned earlier, since Kt typically puts mass on those paths y that resemble Brownian sample-paths, (SE)(a) is like a stochastic differential equation where y is replaced by a Brownian motion. Note that the white noise W is independent of y. Although Kt is not normally atomic, if it were, Kt would keep track of the number of particles alive at time t, as well as the history of each particle’s driving Brownian motions until time t. Dawson et al [2] use a high density limit of a sequence of interacting particle systems, where each of the particles move according to an SDE similar to (SE) (a), to construct a measure-valued diffusion. Earlier, Wang in [24] and [23] constructed special cases of the model of [2]. Using a similar method, the special case where the white noise W is replaced by a Brownian motion B (i.e. setting m = 0 here) was studied initially by Skoulakis and Adler [19] in relation to plankton dynamics. Their model also did not incorporate the spatial interactions between particles that are built into (SE). Before we discuss solutions for (SE), we will need some background material, as well as a framework in which to work. This is provided in Section 2.2. In Section 2.3, we will set up (SE) in a more rigorous manner and then prove the existence and pathwise uniqueness of solutions for a broad range of coefficients σ1,σ2, and b. In Section 2.4, we show that the solutions of (SE) satisfy a martingale problem– an extension of the martingale problems that appears in [2] and [13]. We conclude with a section about various additional properties of the solutions, including the strong Markov and compact support properties. 2.2 Preliminaries For a good reference for much of what follows in this section, consult Chapters II and V of [14]. Let K be an (Ft)-adapted, critical, historical super-Brownian motion on the space (Ω,F,(Ft)t≥0,P). Let C = C(R+,Rd) and C be its Borel σ -field. Also define Ct = {yt : y ∈ C} 13 and Ct = σ(ys,s≤ t,y ∈C) where yt = y·∧t . Define MF(C) to be the space of finite measures on C. For metric spaces E and E ′, take Cc(E,E ′) and Cb(E,E ′) to be the space of compactly supported continuous functions from E to E ′ and the space of bounded continuous functions from E to E ′, respectively. Notation. For a measure µ on a space E and a measurable function f : E→R, let µ( f ) = ∫ f dµ . Note that Kt takes values in MF(C), and so Kt(·) will typically mean integration over the y variable below. We define a martingale problem for K. Let B̂t = B(· ∧ t) be the path-valued process associated with B, taking values in Ct . Then for φ ∈ bC (bounded and C measurable), if s ≤ t let Ps,tφ(y) = Es,y(φ(B̂t)), where the right hand side denotes expectation at time t given that until time s, B̂ follows the path y. We can now introduce the weak generator, Â, of B̂. If φ : R+×C→ R we say φ ∈D(Â) if and only if φ is bounded, continuous and (Ct)-predictable, and for some Âsφ(y) with the same properties as φ , φ(t, B̂)−φ(s, B̂)− ∫ t s Ârφ(B̂)dr, t ≥ s, is a (Ct)-martingale under Ps,y for all s≥ 0,y ∈Cs. Then if m ∈ MF(Rd), we say K satisfies the historical martingale problem, (HMP)m, if and only if K0 = m a.s. and ∀φ ∈D(Â), Mt(φ)≡ Kt(φt)−K0(φ0)− ∫ t 0 Ks(Âsφ)ds is a continuous (Ft)-martingale with 〈M(φ)〉t = ∫ t 0 Ks(φ 2s )ds ∀t ≥ 0, a.s. The definition of Mt(·) can be extended to form an orthogonal martingale mea- sure using the method of Walsh [22]. Denote by P, the σ -field of (Ft)-predictable sets in R+×Ω. If ψ : R+×Ω×C→ R is P×C-measurable and∫ t 0 Ks(ψ2s )ds< ∞ ∀ t ≥ 0, (2.1) then there exists a continuous local martingale Mt(ψ) with quadratic variation 14 given by 〈M(ψ)〉t = ∫ t 0 Ks(ψ2s )ds. If the expectation of the term in (2.1) is finite, then Mt(ψ) is an L2 martingale. In this paper, we will require m(·) ≡ P(K0(·)) to be a finite, positive measure on Rd . Recall that for a measure space (E,E), the universal completion of E, denoted E∗, is given by E∗ = ∩µ Ēµ , where Ēµ denotes the completion of E with respect to the measure µ and the inter- section is over all probability measures µ on E. Definition. Let (Ω̂, F̂, F̂t) = (Ω×C,F×C,Ft ×Ct). Let F̂∗t denote the universal completion of F̂t . If T is a bounded (Ft)-stopping time (denote the set of all such stopping times by Tb), the normalized Campbell measure associated with T is the measure P̂T on (Ω̂, F̂) given by P̂T (A×B) = P(1AKT (B))m(1)−1 for A ∈ F,B ∈ C. We denote sample points in Ω̂ by (ω,y). Hence, under P̂T , ω has law KT (1)dP and conditionally on ω , y has law KT (·)/KT (1). Note that here P̂T is a probability measure, since P̂T (1) = P(KT (1))m(1)−1 = P(K0(1))m(1)−1 = 1 as Kt(1) is Feller’s critical branching diffusion (a uniformly integrable martingale). To avoid carrying constants of m(1)−1 from line to line, we will without loss of generality assume m(1) = 1. The following results will be useful later. The first is Proposition V.2.4 and the second, Proposition V.3.1 of [14]. Proposition 2.2.1. Assume that T ∈ Tb and ψ ∈ F̂T , bounded. Then Kt(ψ) = KT (ψ)+ ∫ t T ∫ ψ(y)dM(s,y) ∀t ≥ T P-a.s. 15 Proposition 2.2.2. If T ∈ Tb then under P̂T , y is an (F̂t)-adapted Brownian motion stopped at T . Suppose that {W (t,ξ ),ξ ∈Rm, t ≥ 0} is an (Ft)-adapted, d-dimensional Brow- nian sheet on (Ω,F,(Ft)t≥0,P) that is independent of K. For a function φ ∈ Cc(Rm,Rn×d), let Wt(φ) = ∫ t 0 ∫ Rm φ(ξ )dW (s,ξ ). This defines the associated white noise process of W , and hence we will identify the two in what follows. The next theorem will be useful in proving that W is also an (F̂t)-adapted P̂T Brownian sheet for each T ∈ Tb. Theorem 2.2.3. Suppose N is a real-valued, square-integrable, (Ft)-adapted mar- tingale that is independent of K. Then for each T ∈ Tb, N is an (F̂t)-adapted martingale under P̂T and 〈N,y j〉t = 0 ∀t ≥ 0 P̂T -a.s ∀ j ≤ d. Proof. Let A ∈ Cs and B ∈ Fs. Then P̂T [(Nt −Ns)1A1B] = P[(Nt −Ns)KT (A)1B] = P[1B1T≤sKT (A)E(Nt −Ns|Fs)]+P[(Nt −Ns)1B1T>sKT (A)] = P [ (Nt −Ns)1B1T>s ( Ks(A)+ ∫ T s ∫ 1AdM(r,y) )] = P [ (Nt −Ns)1B1T>s ( Ks(A)+ ∫ T∧t s ∫ 1AdM(r,y) )] where in the first equality we use the definition of the Campbell measure and in the second that KT (A)1{T≤s} ∈ Fs. For the third equality, we use the fact that Nt is an (Ft)-martingale and Proposition 2.2.1. Finally we condition on Ft and appeal to the Optional Sampling Theorem for the last expression. By conditioning first on Fs and then on FT we can reduce the above as P̂T [(Nt −Ns)1A1B] = P [ (Nt −Ns)1B′ ∫ T∧t s ∫ 1AdM(r,y) ] = P [ (NT∧t −Ns)1B′ ∫ T∧t s ∫ 1AdM(r,y) ] where B′ = B∩{T > s} ∈ Fs. Now, if we show that NtMt(ψ) is an (Ft)-adapted martingale for any bounded function ψ : R+×C → R, then we may apply the 16 Optional Stopping Theorem to show that the above expression reduces to zero. Note that, NtMt(ψ) = ∫ t 0 NsdMs(ψ)+ ∫ t 0 Ms(ψ)dNs+ 〈N,M(ψ)〉t = ∫ t 0 NsdMs(ψ)+ ∫ t 0 Ms(ψ)dNs since 〈N,M(ψ)〉t = 0 by independence of N and M. It also follows from the inde- pendence that the first term on the right side is a martingale. For the second term we use independence and the square integrability of N to conclude that it is a mar- tingale as well. Hence NtMt(ψ) is a martingale and P̂T [(Nt −Ns)1A1B] = 0. This proves the first assertion. From the independence of N and K we have for constant T , a Borel subset A of C(R+,R) and B ∈ C that, P̂T (N ∈ A,y ∈ B) = P(1N∈AKT (1y∈B)) = P(1N∈A)P(KT (1y∈B)) = P̂T (N ∈ A) P̂T (y ∈ B) . Hence, N and y are independent under P̂T . This implies that 〈N,y j〉 = 0, P̂T -a.s. In the general case where T ∈ Tb, independence of N and y under P̂T no longer necessarily holds. Let Z ≡ 〈N,y j〉. Then under P̂T , for u> T fixed, P̂T ( sup s≤T |Zs|∧1 ) = P ( KT ( sup s≤T |Zs|∧1 )) = P ( Ku ( sup s≤T |Zs|∧1 ) − ∫ u T ∫ sup s≤T |Zs|∧1dM(r,y) ) = 0 In the second line, we have noted that sups≤T |Zs| ∧ 1 is bounded, F̂T -measurable and applied Proposition 2.2.1. To get the third line, we use the fact that Z = 0, P̂u- a.s. by the special case considered above and that the second term is a martingale. Hence 〈N,y j〉t = 0 ∀t ≥ 0 P̂T -a.s. ∀T ∈ Tb (note that we have used the property that under P̂T , y = yT a.s. implicitly here). 17 Let Rm denote the Borel σ -algebra of Rm. The following proposition is needed to complete the proof. Proposition 2.2.4. Let M be an orthogonal martingale measure on (Rm,Rm) as defined in Chapter 2 of Walsh [22]. Suppose further that for each A∈R, t 7→Mt(A) is continuous. Then M is a white noise if and only if its covariance measure is given by Lebesgue measure. For the proof, see Proposition 2.10 of Walsh [22]. Corollary 2.2.5. For each T ∈ Tb, {W (t,ξ ),ξ ∈ Rm, t ≥ 0} is an (F̂t)-adapted Brownian sheet under P̂T . Furthermore, W and y are orthogonal under P̂T ; that is, 〈W (φ),y j〉t = 0 ∀φ ∈Cc(Rm,Rd), t ≥ 0 P̂T -a.s. ∀T ∈ Tb for all j ≤ d. Proof. Let Wt(A) = (W 1t (A), . . . ,W d t (A)). We simply need to verify that under P̂T , {W it } are independent white noises. Each W it is an orthogonal martingale measure under P̂T since for disjoint bounded sets A,B ∈ Rm, W it (A) and W it (A)W it (B) are square-integrable (Ft)-martingales with respect to P, hence also (F̂t)-martingales with respect to P̂T by Theorem 2.2.3. The map t 7→W it (A) is continuous since W it (A)(ω,y) defined on Ω̂ is equal to W it (A)(ω) on Ω, for which the analogous map is continuous and P̂T |Ω is absolutely continuous with respect to P. By noting that Theorem 2.2.3 implies W it (A) 2− tν(A) is an (F̂t)-martingale, where ν is the m-dimensional Lebesgue measure, we see that the covariance mea- sure is deterministic. Hence each W it is a white noise process by Proposition 2.2.4. Then by a final use of Theorem 2.2.3, we see that W it (A)W j t (A)− tν(A)δi, j is an (F̂t)-martingale where δi, j is Kronecker’s δ . Therefore W i(A) and W j(A) are inde- pendent Brownian motions (for any A with ν(A)<∞). Hence Wt is a d-dimensional white noise process. The orthogonality of y,W follows from the previous theorem. Note that this corollary will still hold if W is a white noise on a space E×R+ with covariance measure ν × ` where ν is an arbitrary σ -finite measure on E and ` is the one dimensional Lebesgue measure. Indeed the results below all extend to this generality as well. 18 For a matrix A ∈ Rn×d , let ||A||2 = ∑i, j A2i j. Note for a vector x ∈ Rn, we will use |x|2 instead to denote the same quantity. Definition. We say f ∈ D1(n,d) iff f : R+× Ω̂→ Rn×d is (F̂∗t )-predictable and∫ t 0 ‖ f (s,ω,y)‖2ds< ∞ Kt −a.a. y ∀t ≥ 0,P a.s. We say g ∈ D2(m,n,d) iff g : R+×Rm× Ω̂→ Rn×d is (F̂∗t )-predictable and∫ t 0 ∫ Rm ‖g(s,ξ ,ω,y)‖2dξds< ∞ Kt −a.a. y ∀t ≥ 0,P a.s. Definition. If X ,Y : R+× Ω̂→ E, we say X = Y K-a.e. iff X(s,ω,y) = Y (s,ω,y) for all s ≤ t, Kt-a.e. for all t ≥ 0 P-a.s. If E is a metric space we say that X is continuous K-a.e. iff s 7→ X(s,ω,y) is continuous on [0, t] for Kt-a.a. y for all t ≥ 0 P-a.s. Since with respect to each measure P̂T ,T ∈ Tb, {W (s,ξ ),0 ≤ s} is an (F̂s)- adapted Brownian sheet, we may define the stochastic integral of a function f with∫ t 0 ∫ Rd ‖ f (ξ ,s,ω,y)‖2dξds<∞, P̂T -a.s. as in Walsh [22]. However as this integral depends on T , we denote it by P̂T - ∫ t 0 ∫ f (s,ξ ,ω,y)dW (s,ξ ). Similarly, define P̂T - ∫ t 0 g(s,ω,y)dy(s) to be the stochastic integral g with re- spect to y under P̂T . The following part of Proposition V.3.2 of [14] is useful in relating these different stochastic integrals. Proposition 2.2.6. For g ∈ D1(n,d), there exists an (F̂t)-predictable, K-a.e. con- tinuous process I(g, t,ω,y) such that I(g, t ∧T,ω,y) = P̂T - ∫ t 0 g(s,ω,y)dy(s) for all t ≥ 0, P̂T -a.s. for all T ∈ Tb. Moreover, I is unique. That is, if I′ is an (F̂∗t )-predictable process satisfying the above, then I(g,s,ω,y) = I′(g,s,ω,y) K-a.e. We can prove a similar theorem for P̂T - ∫ t 0 ∫ f (s,ξ ,ω,y)dW (s,ξ ), but we will need to recognize that it only holds for t ≤ T as the following example demon- strates. 19 Example 2.2.7. Consider f (s,ξ ,ω,y)= yu(s)η(ξ )1(u,v](s)with η ∈C2c (Rm). Note that under P̂T , y = yT a.s., and hence P̂T - ∫ t 0 ∫ f (s,ξ ,ω,y)dW (s,ξ ) = yT (u)(Wv∧t(η)−Wu∧t(η)) . This expression clearly depends on the stopping time T . If T = T1 < u is fixed then our integral is y(T1)(Wv∧t(η)−Wu∧t(η)), else if T = T2 > u is fixed, then the integral is y(u)(Wv∧t(η)−Wu∧t(η)). If t < T1, then the integrals in both cases agree (and are equal to zero). This shows that even simple functions do not necessarily have unique integrals when integrated with respect to W under the different measures P̂T when t > T . The next theorem is the analogue of Proposition 2.2.6 and shows how the stochastic integrals of W are related under different measures P̂T . Theorem 2.2.8. (a) If f ∈D2(m,n,d), there is anRn-valued (F̂t)-predictable pro- cess J( f , t,ω,y) such that for all T ∈ Tb J( f , t ∧T,ω,y) = P̂T - ∫ t∧T 0 ∫ Rm f (s,ξ ,ω,y)dW (s,ξ ) ∀t ≥ 0, P̂T -a.s. (b) If J′( f ) is an (F̂∗t )-predictable process satisfying (a), then J( f ,s) = J′( f ,s) K-a.e. (c) J( f , t) is continuous in t, K-a.e. (d) (Dominated Convergence) For any N > 0, if fk, f ∈ D2(m,n,d) satisfy lim k→∞ P ( KN (∫ N 0 ∫ Rm ‖ fk(s,ξ )− f (s,ξ )‖2dξds> ε )) = 0,∀ε > 0, then, lim k→∞ P ( sup t≤N Kt ( sup s≤t ‖J( fk,s)− J( f ,s)‖2 > ε )) = 0 ∀ε > 0. 20 (e) For any S ∈ Tb if fk, f ∈ D2(m,n,d) satisfy lim k→∞ P ( KS (∫ S 0 ∫ Rm ‖ fk(s,ξ )− f (s,ξ )‖2dξds )) = 0, then, sup t≤S Kt ( sup s≤t ‖J( fk,s)− J( f ,s)‖2 ) P→ 0 as k→ ∞. The proof of Proposition 2.2.6 can be easily adapted to prove this theorem. We will at times write ∫ t 0 ∫ f (s,ξ ,ω,y)dW (s,ξ ) or Wt( f ) in place of J( f , t,ω,y) where the latter is well defined. Corollary 2.2.9. Let T ∈ Tb. If f ∈ D1(n,d), g ∈ D2(m,n,d) and there exists an (F̂∗t )-predictable process S( f ,g) satisfying S( f ,g, t ∧T,ω,y) = P̂T - ∫ t∧T 0 f (s,y)dy(s)+ P̂T - ∫ t∧T 0 ∫ g(s,ξ )dW (s,ξ ) (2.2) ∀t ≥ 0, P̂T -a.s., then S( f ,g) = I( f )+ J(g),K-a.e. Proof. Define L(t,ω) = ∫ sup s≤t ‖I( f ,s)+ J( f ,s)−S( f ,g,s)‖∧1Kt(dy). (2.3) Note that L is (Ft)-predictable. The proof of this is essentially given in the last half of the proof for Proposition V.3.2(b) of [14]. Assume T is a bounded, predictable stopping time. Then P(L(T,ω)) = P̂T ( sup s≤T ‖I( f ,s)+ J( f ,s)−S( f ,g,s)‖∧1 ) = 0 since S( f ,g,T ∧ s) = P̂T - ∫ s∧T 0 f (s,y)dy(s)+ P̂T - ∫ s∧T 0 ∫ g(s,ξ )dW (s,ξ ), I( f ,s∧T ) = P̂T - ∫ s∧T 0 f (s,ω,y)dy(s) by Proposition 2.2.6 and since J(g,s∧T ) = P̂T - ∫ s∧T 0 ∫ g(s,ξ ,ω,y)dW (s,ξ ) by Theorem 2.2.8(a) above. Then by the Section Theorem, we have that L(t,ω) = 0 ∀t ≥ 0 a.s. 21 Notation. If X(t) = (X1(t), . . . ,Xn(t)) is an Rn-valued process on (Ω̂, F̂) and µ ∈ MF(C), let µ(Xt) = (µ(X1(t)), . . . ,µ(Xn(t))) where µ(Xi(t)) = ∫ Xi(t,ω,y)µ(dy). Also let∫ t 0 ∫ X(s)dM(s,y) = (∫ t 0 ∫ X1(s)dM(s,y), . . . , ∫ t 0 ∫ Xn(s)dM(s,y) ) whenever these integrals are defined. We do the same for stochastic integrals with respect to W . The next theorem is needed in order to prove a version of Itô’s Lemma. Theorem 2.2.10. If f ∈ D2(m,n,d) and sups,ω,y ∫ ‖ f (s,ξ ,ω,y)‖2dξ < ∞ then Kt(J( f , t)) = ∫ t 0 ∫ Ws( f (s))dM(s,y)+ ∫ t 0 ∫ Ks( f (s))dW (s,ξ ) ∀t ≥ 0 P-a.s. (2.4) and each integral on the right is a continuous L2 (Ft)-martingale. Proof. To simplify the notation, assume that n = d = 1. We first show the result for simple functions. Let f (s,ξ ,ω,y) = φ1(ω)φ2(y)φ3(ξ )1(u,v](s) (2.5) where φ1 ∈ bFu,φ2 ∈ bCu,φ3 ∈C∞c (Rm) and 0≤ u< v. Then, Kt (∫ t 0 ∫ f (s,ξ ,ω,y)dW (s,ξ ) ) = Kt (φ1(ω)φ2(y)(Wv∧t(φ3)−Wu∧t(φ3))) = φ1(ω)(Wv∧t(φ3)−Wu∧t(φ3))Kt(φ2) = ∫ t 0 ∫ φ1(ω)φ3(ξ )1(u,v](s)dW (s,ξ )Kt(φ2) where in the second line we make use of the fact that Wt and φ1 depend only on ω . We can now apply the integration by parts formula, while noting that 22 〈W (φ3),K(φ2)〉t = 0 (by independence), to get Kt (∫ t 0 ∫ f (s,ξ )dW (s,ξ ) ) = ∫ t 0 ∫ φ1(ω)φ3(ξ )1(u,v](s)Ks(φ2)dW (s,ξ ) + ∫ t u ∫ s u ∫ φ1(ω)φ3(ξ )1(u,v](r)dW (r,ξ )dKs(φ2) = ∫ t 0 ∫ Ks( f (s,ξ ))dW (s,ξ ) + ∫ t 0 [∫ s 0 ∫ φ1(ω)φ3(ξ )1(u,v](r)dW (r,ξ ) ]∫ φ2(y)dM(s,y) = ∫ t 0 ∫ Ks( f (s))dW (s,ξ )+ ∫ t 0 ∫ [∫ s 0 ∫ f (r,ξ ,ω,y)dW (r,ξ ) ] dM(s,y). Here in the second line we have used Proposition 2.2.1 with T = u and then the fact that the function φ1(ω)φ3(ξ )1(u,v](s) disappears for s < u. The differential∫ φ2(y)dM(s,y) is just dMs(φ2). Note that for these simple functions f , we need only look at the square function of ∫ t 0 ∫ Ks( f (s,ξ ))dW (s,ξ ) to show that it is an L2 martingale. That is, since φ1φ2 is bounded and φ3 ∈C∞c (Rm), P (∫ t 0 ∫ Ks( f (s,ξ ))2dξds ) ≤ cP (∫ t 0 ∫ φ3(ξ )2Ks(1)2dξds ) . This expectation is finite because Kt(1) is Feller’s branching diffusion. To show that ∫ t 0 ∫ J( f ,s)dM(s,y) is an L2 martingale consider P (∫ t 0 Ks ( J( f ,s)2 ) ds ) = ∫ t 0 P ( Ks ( φ 22 )[∫ s 0 ∫ φ1φ31(u,v]dW (r,ξ ) ]2) ds ≤ c ∫ t 0 P ( Ks (1)P [∫ s 0 ∫ (φ1φ31(u,v])2dξdr ]) ds < ∞. Hence the sum of these two quantities is also an L2 martingale. Since Mt(φ2) is continuous, ∫ t 0 ∫ J( f ,s)dM(s,y) is continuous. As ∫ t 0 ∫ Ks( f (s,ξ ))dW (s,ξ ) is con- tinuous so is Kt (∫ t 0 ∫ f (s,ξ ,ω,y)dW (s,ξ ) ) . Suppose the statement of the theorem holds for a sequence of (F̂t)-predictable 23 processes fk and f is an (F̂∗t )-predictable process such that fk → f pointwise, supk ∫ ‖ fk(ξ )‖2dξ ∨ ∫ ‖ f (ξ )‖2dξ < ∞ and lim k→∞ P (∫ ∫ N 0 ∫ ( fk(s,ξ )− f (s,ξ ))2dξdsKN(dy) ) = 0 ∀N ∈ N. (2.6) Since J( f ,s) is a White noise integral with respect to P̂s, consider for each N ∈ N, P(〈M (J( f ,s)− J( fk,s))〉N) = P (∫ N 0 ∫ (J( fk,s)− J( f ,s))2Ks(dy)ds ) = ∫ N 0 P̂s ( (J( fk,s)− J( f ,s))2 ) ds = ∫ N 0 P̂s (∫ s 0 ∫ ( fk(r,ξ )− f (r,ξ ))2dξdr ) ds = ∫ N 0 P̂N (∫ s 0 ∫ ( fk(r,ξ )− f (r,ξ ))2dξdr ) ds where in the second line we have used Fubini’s theorem and the definition of the Campbell measure P̂s. In the third line we make use of the Itô isometry and then finally we use Remark V.2.5 (d) of [14] in the last line. The last expres- sion now goes to zero as k→ ∞ by (2.6) (and Fubini’s theorem). This shows that∫ t 0 ∫ J( f ,s,y)dM(s,y) is a continuous L2 martingale. Now by the Dominated Con- vergence Theorem, ∀t as k→ ∞, P (〈J(Ks( f (s)))− J(Ks( fk(s)))〉t)= P(∫ t 0 ∫ Ks( f (s,ξ )− fk(s,ξ ))2dξds ) → 0, and so ∫ t 0 ∫ Ks( f (s,ξ ))dW (s,ξ ) is a continuous L2 martingale as well. By using Theorem 2.2.8(e) with S = N and (2.6), we get sup t≤N Kt(|J( fk, t)− J( f , t)|) P→ 0 as k→ ∞,∀N ∈ N. Sending k→ ∞ now shows that (2.4) holds for f as well. The rest of the proof proceeds by appealing to a Monotone Class Theorem to pass to the bounded pointwise closure of functions satisfying (2.4). This is exactly 24 what is done in the last part of the proof of Proposition V.3.4 of [14]. Theorem 2.2.11 (Itô’s Lemma). Let Z0 be F̂0-measurable and take values in Rn. Let f ∈ D1(n,d), g ∈ D2(m,n,d), h an Rn-valued (F̂∗t )-predictable process and ψ ∈C1,2b (R+×Rn). Assume∫ t 0 Ks(‖ fs‖2+ |hs|)ds+ ∫ t 0 Ks (∫ ‖g(s,ξ )‖2dξ ) ds< ∞, ∀t a.s., (2.7) and let Zt(ω,y) = Z0(ω,y) (2.8) + ∫ t 0 f (s,ω,y)dy(s)+ ∫ t 0 ∫ g(s,ξ ,ω,y)dW (s,ξ )+ ∫ t 0 h(s,ω,y)ds, then ∫ ψ(t,Zt)Kt(dy) = ∫ ψ(0,Z0)dK0(y) + ∫ t 0 ∫ ψ(s,Zs)dM(s,y)+ ∫ t 0 ∫ Ks(∇ψ(s,Zs)g(s,ξ ))dW (s,ξ ) + ∫ t 0 Ks ( ∂ψ ∂ s (s,Zs)+∇ψ(s,Zs) ·hs+ 12 n ∑ i=1 n ∑ j=1 ψi, j(s,Zs)ai j(s) ) ds, (2.9) where ∇ψ and ψi j are the gradient and second order partial derivatives in the spatial variables and a ≡ f f ∗ + ∫ gg∗(ξ )dξ . The second term on the right is an L2 martingale, the third is a local martingale and the last term on the right has continuous paths with finite variation over compact intervals a.s. Therefore∫ ψ(t,Zt)Kt(dy) is a continuous (Ft)-semimartingale. Proof. For now assume that ‖ f‖ and |h| are bounded and that sups,y ∫ ‖g(s,ξ ,ω,y)‖2dξ < ∞. Let T ∈ Tb and Z̄t = (t,Zt). Using the classical Itô’s Lemma we have, P̂T -a.s., ∀t ≥ 0 25 ψ(Z̄T∧t)−ψ(Z̄0) =∫ T∧t 0 ∇ψ(Z̄s) ·dZs+ ∫ T∧t 0 ∂ψ ∂ s (Z̄s)ds+ 1 2 ∑i, j≤n ∫ T∧t 0 ψi j(Z̄s)d〈Zi,Z j〉s = P̂T - ∫ T∧t 0 ∇ψ(Z̄s) f (s) ·dy(s)+ P̂T - ∫ T∧t 0 ∫ ∇ψ(Z̄s)g(s,ξ ) ·dW (s,ξ ) + ∫ T∧t 0 ( ∇ψ(Z̄s) ·h(s)+ ∂ψ∂ s (Z̄s)+ 1 2 n ∑ i=1 n ∑ j=1 ψi, j(Z̄s)ai j(s) ) ds. Here in the second line we have simply plugged in (2.8), used Corollary 2.2.5 to manage the square function and done some linear algebra. Letting b̃(s) denote the term inside the last integral above, we see that S(t) = ψ(Z̄t)−ψ(Z̄0)− ∫ t 0 b̃(s)ds is an (F̂∗t )-predictable process satisfying (2.2). Hence we apply Corollary 2.2.9 (with f and g replaced by ∇ψ(Z̄) f and ∇ψ(Z̄)g respectively) to get ψ(Z̄t) = ψ(Z̄0)+ I(∇ψ(Z̄) f , t)+ J(∇ψ(Z̄)g, t)+ ∫ t 0 b̃(s)ds. (2.10) Since |b̃|,‖∇ψ(Z̄) f‖ and ∫ ‖∇ψ(Z̄)g(ξ )‖2dξ are all bounded and (F̂∗t )-predictable, we can apply Theorem 2.2.10 above, Proposition V.3.4 and Re- 26 marks V.2.5 of [14] to get P-a.s. for all t ≥ 0,∫ ψ(Z̄t)Kt(dy) = ∫ ψ(Z̄0)K0(dy)+ ∫ t 0 ∫ ψ(Z̄0)dM(s,y) + ∫ t 0 ∫ I(∇ψ(Z̄) f ,s)dM(s,y)+ ∫ t 0 ∫ J(∇ψ(Z̄)g,s)dM(s,y) + ∫ t 0 ∫ [∫ s 0 b̃(r)dr ] dM(s,y)+ ∫ t 0 ∫ Ks(∇ψ(Z̄s)g(s,ξ ))dW (s,ξ ) + ∫ t 0 Ks(b̃(s))ds = ∫ ψ(Z̄0)K0(dy)+ ∫ t 0 ∫ ψ(Z̄s)dM(s,y) + ∫ t 0 ∫ Ks(∇ψ(Z̄s)g(s,ξ ))dW (s,ξ )+ ∫ t 0 Ks(b̃(s))ds. In the second equality, we used (2.10) to simplify. This gives the result in this case. Now assume that f ,g and h satisfy (2.7). We may choose f k,gk and hk bounded and (F̂∗t )-predictable and such that ∫ ‖gk(ξ )‖2dξ is bounded and f k → f ,gk → g and hk → h pointwise (by, for example, truncating f ,g and h) with ‖ f k‖ ≤ ‖ f‖,‖gk‖ ≤ ‖g‖ and |hk| ≤ |h|. Therefore, by the Dominated Convergence Theo- rem,∫ t 0 Ks ( ‖ f ks − fs‖2+ |hks −hs|+ ∫ ‖gks −gs‖2dξ ) ds→ 0 ∀t ≥ 0 a.s. (2.11) Using (2.7) choose Sn ∈ Tb, Sn ↑ ∞, a.s. such that∫ Sn 0 Ks (‖ fs‖2+ |hs|)ds+∫ Sn 0 Ks (∫ ‖gs‖2dξ ) ds≤ n a.s. Let Zk be as in (2.8) except with f ,g,h replaced by f k,gk,hk respectively. Using Lemma V.3.3(b) of [14] we get that supt≤Sn Kt( ∫ t 0 |hs|ds) < ∞ for all n, a.s. and hence Zt(ω,y) is well-defined Kt-a.a. y for all t ≥ 0 a.s. The same lemma shows that sup t≤Sn Kt (∫ t 0 |hks −hs|ds ) P→0 as k→ ∞ ∀n. (2.12) 27 Using Proposition V.3.2(e) and Lemma V.3.3(b) of [14] gives sup t≤Sn Kt ( sup s≤t ‖I( f k,s)− I( f ,s)‖2 ) P→ 0 as k→ ∞ ∀n. (2.13) Similarly, using Theorem 2.2.8 above and Lemma V.3.3(b) of [14] gives sup t≤Sn Kt ( sup s≤t ‖J(gk,s)− J(g,s)‖2 ) P→ 0 as k→ ∞ ∀n. (2.14) Together, (2.12), (2.13) and (2.14) give sup t≤T Kt ( sup s≤t |Zk(s)−Z(s)| ) P→0 as k→ ∞ ∀T > 0. (2.15) Since we have that (2.9) holds for (Zk, f k,gk,hk) in place of (Z, f ,g,h), the bound- edness of ψ and its derivatives together with (2.11) and (2.15) lets us send k→ ∞ and derive the result using the Dominated Convergence Theorem. Note that the proofs of Theorem 2.2.10 above and Proposition V.3.4 of [14] imply that the second term of (2.9) is an L2 martingale and the third is a local martingale. The condition (2.7) can be used to show that the last term of (2.9) has finite variation on bounded intervals a.s. Corollary 2.2.12. If f ,g,h and Z are as in Theorem 2.2.11, then Xt(A) = ∫ 1(Zt(ω,y) ∈ A)Kt(dy) defines an a.s. continuous (Ft)-predictable MF(Rn)-valued process. The proof for this corollary is the same as for Corollary V.3.6 of [14]. 2.3 The strong equation In this section, let K be as in Section 2.2 and let ν be the law of K0. Let σ1 : MF(Rd)×Rd → Rd×d , σ2 : MF(Rd)×Rd ×Rm → Rd×d , b : MF(Rd)×Rd → Rd and Z0 : Rd → Rd be Borel maps. If ∫ t 0 σ1(Xs,Zs)dy(s) is the integral I and∫ t 0 ∫ σ2(Xs,Zs,ξ )dW (s,ξ ) is the integral J discussed in Section 2.2 above, then the 28 precise interpretation of (SE) is as follows: (a) Zt(ω,y) = Z0(y0)+ ∫ t 0 σ1(Xs,Zs)dy(s) (SE)Z0,K + ∫ t 0 ∫ σ2(Xs,Zs,ξ )dW (s,ξ )+ ∫ t 0 b(Xs,Zs)ds K-a.e. (b) Xt(ω)(A) = ∫ 1(Zt(ω,y) ∈ A)Kt(dy) ∀A ∈B(Rd),∀t ≥ 0 a.s. (X ,Z) is a solution of (SE)Z0,K iff Z is an (F̂ ∗ t )-predictable Rd-valued process and X is an (Ft)-predictable MF(Rd)-valued process such that (SE)Z0,K holds. Before we start discussing the existence of a solution of (SE)Z0,K , we must impose some conditions on the problem to make it more tractable. Let Lip1 = {φ : Rd → R : ‖φ‖∞ ≤ 1, |φ(x)− φ(z)| ≤ |x− z| ∀x,z ∈ Rd} and for µ,ν ∈ MF(Rd), denote the Vasserstein metric on MF(Rd) by d(µ,ν) = sup φ∈Lip1 |µ(φ)−ν(φ)|. Let σ1,σ2 and b be such that there is an increasing function L : R+→ R+ with Lip (a) ‖σ1(µ,z)−σ1(µ ′,z′)‖+ |b(µ,z)−b(µ ′,z′)| ≤ L(µ(1)∨µ ′(1))[d(µ,µ ′)+ |z− z′|] ∀µ,µ ′ ∈MF(Rd),z,z′ ∈ Rd (b) ∫ ‖σ2(µ,z,ξ )−σ2(µ ′,z′,ξ )‖2dξ ≤ L(µ(1)∨µ ′(1))[d(µ,µ ′)2+ |z− z′|2] (c) sup z (‖σ1(0,z)‖+ |b(0,z)|)< ∞ and sup z ∫ ‖σ2(0,z,ξ )‖2dξ < ∞. The following Lemma follows easily from the Lipschitz type conditions above. Lemma 2.3.1. There exists a non-decreasing function α : R+→ R+ such that sup z [ ‖σ1(µ,z)‖+ ∫ ‖σ2(µ,z,ξ )‖2dξ + |b(µ,z)| ] ≤ α(µ(1)) (2.16) Define TN = inf{t : Kt(1)≥ N}∧N and 29 S1 = {X : R+×Ω→MF(Rd) : X is (Ft)-predictable, a.s. continuous and Xt(1)≤ N ∀t < TN ∀N ∈ N} S2 = {Z : R+× Ω̂→ Rd : Z is (F̂∗t )-predictable and continuous K-a.e.} S = S1×S2. Let Φ2(X ,Z)(t,y)≡ Z̃t(y) = Z0(y0)+ ∫ t 0 σ1(Xs,Zs)dys+ ∫ t 0 ∫ σ2(Xs,Zs,ξ )dW (s,ξ )+ ∫ t 0 b(Xs,Zs)ds Φ1(X ,Z)(t)(·)≡ X̃t(·) = ∫ 1(Z̃t(ω,y) ∈ ·)Kt(dy) and let Φ= (Φ1,Φ2). Then note that Φ : S→ S by Corollary 2.2.12. Lemma 2.3.2. Let (X̃ i, Z̃i) = Φ(X i,Zi) for i = 1,2. If there exist universal con- stants cN such that for any stopping time T ≤ TN , P̂T ( sup s≤t∧T |Z̃1(s)− Z̃2(s)|2∧1 ) ≤ cN ∫ |Z̃10 − Z̃20 |2∧1dm (2.17) + cN [ P (∫ t∧T 0 d(X1s ,X 2 s ) 2∧1ds ) + P̂T (∫ t∧T 0 |Z̃1(s)− Z̃2(s)|2∧1ds )] , ∀N ∈ N, then there is a pathwise unique solution (X ,Z) to (SE)Z0,K . In particular, X is unique up to P-null sets and Z is unique K-a.e. Also, the map t 7→ Xt is a.s. continuous in t. Proof. The proof of this lemma is essentially provided in the proof of Theorem V.4.1(a) of [14]. To see this, first set K1 = K2 in that proof, and then note that inequality (2.17) above is the same as inequality V.4.9 there, and from there on follow the proof substituting (2.17) for V.4.9 where it appears. The idea is to perform a contraction argument on the complete metric space (S,d0), where d0 is an appropriately chosen metric that depends on the sequence {cN}. One has to prove that Φ is a contraction on S. Theorem 2.3.3. If (Lip) holds, then there exists a pathwise unique solution (X ,Z) 30 to (SE)Z0,K . In particular, X is unique up to P-null sets and Z is unique K-a.e. Also, the map t 7→ Xt is a.s. continuous in t. Proof. We will simply verify that the hypotheses in Lemma 2.3.2 hold. Let TN be as above and T ∈ Tb with T ≤ TN . Let (X̃ i, Z̃i) be as above, for i = 1,2. Using first Doob’s strong L2 inequality and then successive applications of the Cauchy- Schwartz inequality and the fact that TN ≤ N, we have P̂T ( sup s≤t∧T |Z̃1(s)− Z̃2(s)|2∧1 ) ≤ cP̂T ( |Z10 −Z20 |2∧1+ ∫ t∧T 0 ∫ ‖σ2(X1s ,Z1s ,ξ )−σ2(X2s ,Z2s ,ξ )‖2dξds ) + cP̂T [∫ t∧T 0 ‖σ1(X1s ,Z1s )−σ1(X2s ,Z2s )‖2+N|b(X1s ,Z1s )−b(X2s ,Z2s )|2ds ] ≤ cP̂T (|Z10 −Z20 |2∧1) + cNP̂T (∫ t∧T 0 L(X1s (1)∨X2s (1))2(d(X1s ,X2s )+ |Z1s −Z2s |)2 ∧ (α(X1s (1))2+α(X2s (1))2)ds ) ≤ cP̂T [ |Z10 −Z20 |2∧1 +NL(N)2 ∫ t∧T 0 (d(X1s ,X 2 s )+ |Z1s −Z2s |)2∧α(N)2ds ] , (2.18) where we have used (Lip) (a), (b) and Lemma 2.3.1 for the third inequality. For the fourth inequality we use the fact that T ≤ N and Corollary 2.2.12 (to conclude that X is(1) ≤ N for s ≤ T ) and that α and L are non-decreasing. Note that in the above, all the constants (not depending on N) are collected under the term c, and this term then varies from line to line. Also, if T = 0, then KT (1) may exceed N, but the integral in (2.18) above is then zero. If T > 0 then TN > 0 and so 31 KT (1)≤ supt≤T Kt(1)≤ N. Hence we have P̂T ( sup s≤t∧T |Z̃1(s)− Z̃2(s)|2∧1 ) ≤ c ∫ |Z10 −Z20 |2dm+ cN2L(N)2α(N)2P (∫ t∧T 0 d(X1s ,X 2 s ) 2∧1ds ) + cNL(N)2α(N)2P̂T (∫ t∧T 0 |Z1s −Z2s |)2∧1ds ) , where we have used Proposition 2.2.1 to handle the first term and then factored out the α(N)2 term inside the integral in (2.18). Letting cN = N2L(N)2α(N)2 and invoking Lemma 2.3.2 completes the proof. Examples of functions σ1,b satisfying (Lip) (a) can be found in Section V.1 of [14]. The following remark gives some broad classes of coefficients σ2 for which (Lip)(b) holds. Remark 2.3.4. Suppose there exists a finite constant c0 such that for all ξ , ‖σ2(µ,z,ξ )−σ2(µ ′,z′,ξ )‖ ≤ c0(d(µ,µ ′)+ |z− z′|) and sup z ∫ ‖σ2(0,z,ξ )‖2dξ < ∞. Additionally, suppose one of the following properties hold: (i) σ2(µ,z, ·) has compact support for each µ,z and satisfying |suppσ2(µ,z, ·)| ≤ L(µ(1)) for all z for some increasing function L : R+ → R+ (where |A| represents the m-dimensional volume of A⊂ Rm). (ii) σ2(µ,z,ξ ) has no measure dependence and has first and second derivatives in z. Also, there exists M > 0 such that supz ‖∂α(σ2)(z, ·)‖2 < M for all multi-indices α up to order 2. (iii) σ2(µ,z,ξ ) has no measure dependence, is continuous, integrable in ξ for each z, and has uniformly bounded first and second derivatives in z. Also assume that there exists M > 0 such that for all i ≤ d, supz ‖ ∂∂ ziσ2(z, ·)‖1 <M. 32 (iv) σ2(µ,z,ξ ) = ∫ f (z,x,ξ )dµn(x) where x∈Rnd and f :Rd+nd+m→Rd×d such that f is bounded, twice differentiable in x and z and there exists M > 0 with supz,x ‖∂α f (z,x, ·)‖2 <M for all multi-indices α with |α| ≤ 2. Additionally assume that there exists an L2 function c : Rm→ R+ such that for all z, the function f (z, ·,ξ ) is Lipschitz with constant c(ξ ). (v) σ2(µ,z,ξ ) = f (µ(φ1), . . . ,µ(φn),z,ξ ) where f : Rn+d+m → Rd×d and such that f ((x,z), ·) satisfies the conditions on σ2 in either (ii) or (iii) above and each φi bounded and measurable. Then (Lip) (b) holds. Proof. The proof of (Lip) (b) under case (i) is not hard (just use the Lipschitz property followed by the compact support property). Assume, for case (ii) that gξ (z) = ‖σ2(z,ξ )−σ2(z0,ξ )‖2. Since σ2 is twice differentiable, so is gξ . Note that for each y, gξ has a minimum when z = z0 and that gξ (z0) = 0. Hence, by looking at the Taylor expansion we get gξ (z) = ∑ |α|=2 2 α! (z− z0)α ∫ 1 0 (1− t)∂α(gξ )(z0+ t(z− z0))dt and, after letting z(t) = z0+ t(z− z0) we have∫ gξ (z)dξ ≤ c|z− z0|2 ∑ α=2 ∫ ∫ 1 0 (1− t)‖∂α(gξ )(z(t))‖dtdξ ≤ c1|z− z0|2 ∑ α=2 ∫ 1 0 ∫ ∑ i, j≤d |∂α ( σ i j2 (z(t),ξ )−σ i j2 (z0,ξ ) )2 |dξdt ≤ c2M2|z− z0|2 by the uniform L2 bound on the derivatives up to order 2 of σ2. This proves the lemma under this case. For (iii) first define the function h(z) = ∫ ‖σ2(z,ξ )−σ2(z0,ξ )‖2dξ . Then note that h(z0) = 0 and that this is a minimum for h. Therefore, since the first partials 33 of h vanish at z0 h(z1)≤ |∇zh(z0) · (z1− z0)|+ csup z ‖∇2z h(z)‖|z1− z0|2 ≤C|z1− z0|2 by the boundedness of the second derivatives of h (which follows from the assump- tions on σ2). This gives us the result in this case. For case (iv), consider σ2(µ1,z1,ξ )−σ2(µ2,z2,ξ ) = ∫ Rnd f (z1,x,ξ )d(µn1 −µn2 )(x)− ∫ Rnd ( f (z2,x,ξ )− f (z1,x,ξ ))dµn2 (x) and hence through applications of Cauchy-Schwartz and Fubini, we get∫ ‖σ2(µ1,z1,ξ )−σ2(µ2,z2,ξ )‖2dξ ≤ 2 ∫ c(ξ )2d(µn1 ,µ n 2 ) 2dξ +2µ2(1)2n ∫ ∫ ‖ f (z2,x,ξ )− f (z1,x,ξ )‖2dξdµn2 (x) since f (z, ·,ξ )/c(ξ ) ∈ Lip1(nd) for each pair z,ξ by the assumptions on f . Then by using the same proof as in part (iii) to handle the second term, the fact that c(ξ ) is L2, and the definition of Lip1(nd) we get∫ ‖σ2(µ1,z1,ξ )−σ2(µ2,z2,ξ )‖2dξ ≤Cd(µ1,µ2)n+C(µ1(1)µ2(1))2n|z1− z2|2 ≤ L(µ1(1)∨µ2(1))[d(µ1,µ2)2+ |z1− z2|2] by choosing L(x) =Cxn−1+Cx2n. This gives (Lip) (b). The proof for case (v) follows easily from the proofs of either (ii) or (iii) and the definition of the Vasserstein metric. 34 2.4 A martingale problem We will now show that X satisfies a certain martingale problem; a generalized ver- sion of the martingale problem satisfied by the process constructed in [2] and an ex- tension to the one considered in [13]. First we need to show that any solution (X ,Z) to (SE)Z0,K satisfies the following martingale problem (MP): X0 = µ ∈MF(Rd) and for any φ ∈C2b(Rd), MXt (φ)≡ Xt(φ)−µ(φ)− ∫ t 0 Xs(LXsφ)ds (2.19) is a continuous martingale with square function 〈MX(φ)〉t (2.20) = ∫ t 0 Xs(φ 2)ds+ ∫ t 0 ∫ ∫ ∇φ(z1)ρXs(z1,z2)∇φ ∗(z2)dXs(z1)dXs(z2)ds where for ν ∈MF(Rd), Lνφ(z) = d ∑ i=1 bi(ν ,z)φi(z)+ 1 2 d ∑ i, j=1 ai j(ν ,z)φi j(z) with a = σ1σ∗1 + ∫ σ2σ∗2 (ξ )dξ (2.21) and ρν is the d×d matrix given by ρν(z1,z2) = ∫ Rm σ2(ν ,z1,ξ )σ∗2 (ν ,z2,ξ )dξ . Lemma 2.4.1. Any solution (X ,Z) of (SE)Z0,K satisfies (MP) with X0(·) = ∫ 1(Z0 ∈ ·)K0(dy). Proof. The proof follows directly from Theorem 2.2.11 and (SE)Z0,K . First we note that for φ : Rd → R bounded and measurable, (SE)Z0,K implies that Xt(φ) =∫ φ(Zt(y))Kt(dy). Then from Itô’s Lemma (Theorem 2.2.11), for ψ ∈C2b(Rd) we 35 get ∫ ψ(Zt)Kt(dy) = ∫ ψ(Z0)dK0(y)+Nt + ∫ t 0 Ks ( ∇ψ(Zs) ·b(Xs,Zs)+ 12 n ∑ i, j=1 ψi, j(Zs)ai j(Xs,Zs) ) ds, where Nt is a martingale. This becomes, by the boundedness of ψ and its partials, Xt(ψ) = X0(ψ)+Nt + ∫ t 0 ∫ LXsψ(z)Xs(dz)ds. Since Nt = ∫ t 0 ∫ ψ(Zs)dM(s,y)+ ∫ t 0 ∫ Ks(∇ψ(Zs)σ2(Xs,Zs,ξ ))dW (s,ξ ), by the orthogonality of M and W we have 〈N〉t = ∫ t 0 Xs(ψ2)ds + ∫ t 0 ∫ (∫ ∇ψσ2(Xs,z,ξ )dXs(z) )(∫ ∇ψσ2(Xs,z,ξ )dXs(z) )∗ dξds, which can be rewritten in the form of (2.20). Definition. For a function f : Rnd → R, define Fn, f : MF(Rd)→ R by Fn, f (µ) =∫ f dµn. It can be shown, by induction and using Lemma 2.4.1, that for f = ⊗n i=1 φi, where each φi ∈C2b(Rd), that Nt(n, f ) = Fn, f (Xt)−Fn, f (X0)− ∫ t 0 L̂Fn, f (Xs)ds (2.22) 36 is a martingale, where L̂Fn, f (µ) = n ∑ i=1 µ(Lφi) ( ∏ j 6=i µ(φ j) ) (2.23) + 1 2 n ∑ i, j=1 i 6= j ( ∏ k 6=i, j µ(φk) )[ µ(φiφ j)+ ∫ Rm µ(∇φiσ2(ξ )) ·µ(∇φ jσ2(ξ ))dξ ] . We have suppressed the measure dependence of the generator L here and will do so below as well. Definition. If f ∈B(Rnd) (i.e. f a real valued, bounded Borel measurable function on Rnd) then we can define the operator Φni, j : B(Rnd)→ B(R(n−1)d) acting on f as: Φni, j f (z1, . . . ,zn−1) = f (z1, . . . ,zn−1, . . . ,zn−1, . . . ,zn−2) where zn−1 is inserted into the vector (z1, . . . ,zn−2) such that it appears at the ith and jth spots. We can now rewrite (2.23) in terms of f as follows: L̂Fn, f (µ) = n ∑ i=1 ∫ Li f dµn+ 1 2 n ∑ i, j=1 i 6= j ∫ Φni, j( f )dµ n−1 (2.24) + 1 2 n ∑ i, j=1 i 6= j d ∑ k,l=1 ∫ ρk,lµ (zi,z j) ∂ 2 ∂ zik∂ z jl f (z)dµn(z). Here z = (z1, . . . ,zn) and Li f (z1, . . . ,zn) = L fz1,...,zi−1,zi+1,...,zn(zi) (ie, applying L to a function of just zi and fixing the other coordinates) where L is as in (2.21). Note that we have suppressed the dependence of Li on ν , as with L. By Lemma 2.4.1 37 and an inductive argument, one can see that Nt(n, f ) = ∫ t 0 ∫ n ∑ i=1 ( ∏ j 6=i Xs(φ j) ) Xs(∇φiσ2(ξ ))dW (s,ξ ) (2.25) + ∫ t 0 ∫ n ∑ i=1 ( ∏ j 6=i Xs(φ j) ) φi(Zs)dM(s,y) = n ∑ i=1 ∫ t 0 ∫ [∫ ∇zi f (z)σ2(zi,ξ )X n s (dz) ] dW (s,ξ ) (2.26) + n ∑ i=1 ∫ t 0 ∫ [∫ f (Zs(y1), . . . ,Zs(yn))Kn−1s (dy i) ] dM(s,yi) where yi = (y1, . . . ,yi−1,yi+1, . . . ,yn). Then a calculation gives 〈N(n, f )〉t = n ∑ i, j=1 ∫ t 0 [∫ ∇i f (z1)ρXs(zi,zn+ j)(∇n+ j f (z2)) ∗X2ns (dz) ] ds (2.27) + n ∑ i, j=1 ∫ t 0 ∫ [∫ Φ2ni, j+n( f ⊗ f )X2n−1s (dz2n) ] ds where z = (z1,z2) with z1 = (z1, . . . ,zn),z2 = (zn+1, . . . ,z2n) and where z2n = (z1, . . . ,z2n−1). Theorem 2.4.2. For any function f ∈C2b(Rnd), Nt(n, f ) = Fn, f (Xt)−Fn, f (X0)− ∫ t 0 L̂Fn, f (Xs)ds (2.28) is a martingale, where L̂ is given by (2.24). Proof. Let Hn = { M ∑ i=1 ai nd⊗ j=1 φi j : φi j ∈C∞c (R),ai ∈ R,M ∈ N } . We will first use functions in Hn to prove that for f ∈ C∞c (Rnd), (2.28) is a mar- tingale and then boost that up to include functions in C2c (Rnd) and finally those in C2b(Rnd). By the Stone-Weierstrass theorem, for any f ∈ C∞c (Rnd), there is a se- quence gk ∈Hn such that gk→ ∂α f uniformly, for α = (2, . . . ,2). Then, by inte- 38 grating in the various components, we obtain a sequence fk such that ∂α fk→ ∂α f uniformly for |α| ≤ 2. This in turn gives point-wise convergence of Fn, fk to Fn, f and L̂Fn, fk to L̂Fn, f on MF(Rd) (using the Dominated Convergence Theorem). Now let TN = inf{t : Kt(1)≥ N}∧N as above. Then for each k, l, P ( sup s≤t∧TN |Ns(n, fk)−Ns(n, fl)|2 ) ≤ cP(〈N(n, fk)−N(n, fl)〉t∧TN ) and hence by sending l→ ∞ and using the Dominated Convergence Theorem, we conclude that P ( sup s≤t∧TN |Ns(n, fk)−Ns(n, f )|2 ) ≤ cP(〈N(n, fk)−N(n, f )〉t∧TN ) → 0 as k → ∞, by the Dominated Convergence Theorem and Lemma 2.3.1. Hence Nt(n, f ) is a continuous local martingale. Since f and its derivatives up to order 2 are bounded and (Lip) (c) holds, we have by (2.27) that P(〈N(n, f )〉t)≤ cP (∫ t 0 Xs(1)2n+Xs(1)2n−1ds ) . The right side is finite by the boundedness of moments of the mass process of super-Brownian motion. Therefore Nt(n, f ) is a martingale. Now let f ∈C2c (Rnd). Using approximate identities we can construct functions fk ∈C∞c (Rnd) such that ∂α fk→ ∂α f uniformly for all α with |α| ≤ 2. Repeating the above argument again and using the fact that each Nt( fk) is a martingale shows that Nt( f ) is a continuous martingale. Finally, suppose f ∈C2b(Rnd). Let ηk ∈C∞c (Rnd) such that ηk = 1 on B(0,k), ηk = 0 on B(0,k+1)c and ‖∂αηk‖< 2 for each multi-index with |α| ≤ 2. Letting fk = fηk, we see that ∂α fk→ ∂α f pointwise for each |α| ≤ 2. Here B(0,k) denotes the open ball of radius k about the origin. This gives L̂Fn, fk → L̂Fn, f pointwise (by the Dominated Convergence Theo- rem). Then by noting that each Nt( fk) is a continuous martingale and using the 39 method in the first part of this proof shows that Nt( f ) is a continuous martin- gale. To see that the above martingale problem reduces to the one found in [2] in their setting, let m = d = 1, b = 0, σ2(µ,z,ξ ) = h(ξ − z) where h is square-integrable and σ1(µ,z) = σ1(z) is Lipschitz.. Additionally, suppose that η(z) ≡ ∫ R h(ξ − z)h(ξ )dξ is twice continuously differentiable with η ′ and η ′′ bounded. Then one can easily check that these coefficients satisfy (Lip) (use Taylor’s theorem on η to show (Lip)(b)). To get the form of the infinitesimal generator, we use (2.24) : L̂Fn, f (µ) = 1 2 n ∑ i=1 ∫ (σ21 (zi)+η(0)) fii(z)dµ n(z)+ 1 2 n ∑ i, j=1 i 6= j ∫ ρ(zi,z j) fi j(z)dµn(z) + 1 2 n ∑ i, j=1 i 6= j ∫ Φni, j( f )dµ n−1 which is equal to the sum of the generators (2.1) and (2.2) of [2], since ρ(zi,z j) = η(zi− z j), if we allow their branching rate σ to be constant at 1. In [2], uniqueness in law of the martingale problem is established. Hence in this particular setting, our solution is both unique in law and pathwise unique and therefore is a version of the process constructed in [2]. 2.5 Additional properties of solutions We will start off with a stability result for solutions of (SE)Z0,K , and then use it to prove the strong Markov property. We conclude with a proof of the compact support property for the solutions of (SE)Z0,K . Recall that d denotes the Vasserstein metric on MF(Rd). Theorem 2.5.1. Assume (Lip) holds. Let K1 ≤ K2 ≡ K be (Ft)-historical super- Brownian motions with mi(·)≡E(Ki0(·))∈MF(Rd), and let Zi0 :Rd→Rd be Borel maps, for i = 1,2. Let TN = inf{t : Kt(1)≥ N}∧N. There are universal constants 40 {cN ,N ∈ N} so that if (X i,Zi) is the unique solution of (SE)Zi0,Ki , then P (∫ TN 0 sup r≤s d(X1(r),X2(r))2ds ) ≤ cN (∫ |Z10 −Z20 |2∧1dm2+m2(1)−m1(1) ) The proof of this theorem essentially follows from the proof of Theorem V.4.2 of [14]. Note that before we can follow that proof, we will need to show that the condition (2.17) of Lemma 2.3.2 above holds under (Lip), where (X̃ i, Z̃i) = Φi(X i,Zi) where each Φi depends on different historical super-Brownian motions Ki, i = 1,2 and K1 ≤ K2 (ie, the Radon-Nikodym derivative dK1dK2 ≤ 1). The compu- tation showing this is almost exactly like the one in the proof of Theorem 2.3.3. We now construct a canonical space upon which we can define a white noise and an independent historical super-Brownian motion. Let K be an (Ft)-adapted historical super-Brownian motion on Ω̄ = (Ω,F,(Ft)t≥0,Qν). Let (X ,Z) be the unique solution to (SE)Z0,K . Recall that X is (Ft)-predictable, and Z is (F̂ ∗ t )- predictable. Denote by ΩX the space of continuous paths in MF(Rd) and FX its Borel σ -field. Let ΩK = {H· ∈C([0,∞),MF(C)) : Ht ∈MtF(C)∀t ≥ 0} where MtF(C) = {µ ∈ MF(C) : µ(A) = µ (A∩{y ∈C : y = yt})∀A ∈ C}. The historical super-Brownian motion K has sample paths in ΩK . Let FK be the Borel σ -field of ΩK and let FKt = ∞⋂ n=1 σ(Hr : 0≤ r ≤ t+1/n). To treat W we will follow the method of [12]. Let QW be the law of the white noise W. For functions f ∈C(Rd), define ‖ f‖λ = sup x∈Rd | f (x)|e−λ |x|, and let Ctem = { f ∈ C(Rm) : ‖ f‖λ < ∞ for all λ > 0}. We let Ctem be endowed with the topology generated by the family of norms ‖ · ‖λ ,λ > 0. Hence, Ctem can be thought of as a metric space with metric d given by d( f ,g) = ∑∞k=1 2−k(‖ f − g‖1/k ∧ 1). It turns out that (Ctem,d) is a Polish space. Let W = (W 1, . . . ,W d) be 41 a d-dimensional white noise, as in Section 2.2. We can now identify each white noise W i with its associated Brownian sheet and say each W i is a process with sample paths in C([0,∞),Ctem). Hence we will say that W has sample paths in ΩW ≡ C([0,∞),Cdtem). Endow ΩW with the topology of uniform convergence on compacts. Let W̃ : [0,∞)×ΩW →Cdtem be the coordinate projection map, and let FWt = ∞⋂ n=1 σ(W̃s : s≤ t+1/n). Then (H,W̃ ) is a Borel strong Markov process on (ΩK,W ,H,Ht ,Qν ,W ) where ΩK,W = ΩK ×ΩW ,Qν ,W = Qν ×QW and for t ≥ 0, Ht is given by σ((H,W̃ )(s) : s≤ t) with the Qν ,W -null sets thrown in. We will now treat H and W̃ as random processes on the space ΩK,W with H(t,ω,α) = Ht(ω) and W̃ (t,ω,α) = αt . Under Qν ,W , H and W̃ are independent by definition and have laws Qν and QW respectively. Recall from the definition in Section 2.2 that Ω̂K,W = ΩK,W ×C and similarly Ĥt =Ht ×Ct with universal completion Ĥ∗t . Statement (a) below shows that (SE)Z0,K has a strong solution, whereas (b) shows that there is continuity of the laws of the solutions in the initial condition. Theorem 2.5.2. Let (X ,Z) be the unique solution of (SE)Z0,K on Ω̄. Then the following holds: (a) There are (Ht)-predictable and (Ĥ∗t )-predictable maps X̃ : R+ ×ΩK,W → MF(Rd) and Z̃ : R+× Ω̂K,W → Rd , respectively, depending only on (Z0,ν), and such that( X(t,ω),Z(t,ω,y) ) = ( X̃ ( t,K(ω),W (ω) ) , Z̃ ( t,K(ω),W (ω),y )) (2.29) gives the unique solution of (SE)Z0,K . (b) There is a continuous map from X0 7→ P′X0 from MF(Rd) to M1(ΩX) such that if (X ,Z) is a solution of (SE)Z0,K on some filtered space Ω̄ then P(X ∈ ·) = ∫ P′X0(ω)(·)dP(ω). (2.30) 42 (c) If T is an a.s. finite (Ft)-stopping time, then for all A ∈ FX P(X(T + ·) ∈ A|FT )(ω) = P′XT (ω)(A),P-a.s. Proof. The proof of (a) is an analogue of the proof of Theorem V.4.1 (b) of [14]. For (a), let (X̃ , Z̃) be the unique solution of (SE)Z0,K where (H,W̃ ) is the canon- ical process on Ω̄′ ≡ (ΩK,W ,H,Ht ,Qν ,W ). It is clear from Theorem 2.3.3 that (X̃ , Z̃) depends only on Z0 and ν , and satisfies the desired predictability condi- tions. To show (2.29), we need only show that the latter process solves (SE)Z0,K on Ω̄. Let I′( f , t,H,W,y),J′(g, t,H,W,y) and f ∈D′1(d,d),g ∈D′2(m,d,d) denote the sample path stochastic integral and the white noise integral respectively from Sec- tion 2.2 on Ω̄′. Similarly define I( f , t),J(g, t), f ∈ D1,g ∈ D2 to be the stochastic integrals in Section 2.2 on Ω̄. We claim that if f ∈ D′1, then f ◦ (K,W )(t,ω,y) ≡ f (t,K(ω),W (ω),y) ∈ D1 and I( f ◦ (K,W )) = I′( f )◦ (K,W ),K-a.e. (2.31) By the definition of D′1 and D1 and since K and H have the same laws, we get the first implication. Equation (2.31) clearly holds for simple functions. Using Proposition V.3.2 (d) of [14] allows us to show that it holds for all f ∈ D′1. The situation with the white noise integral J is similar in that if g∈D′2, then J◦(K,W )∈ D2 and J(g◦ (K,W )) = J′(g)◦ (K,W ),K-a.e.. This can again be shown by first starting at simple functions and then bootstrap- ping up using Theorem 2.2.8. To show the result, we can now simply replace (H,W̃ ) with (K(ω),W (ω)) in (SE)Z0,H and get that (X̃ ◦(K,W ), Z̃ ◦(K,W )) solves (SE)Z0,K on Ω̄. The proof of (b) is also a straight-forward analogue of the proof of Theorem V.4.1 (c) in [14]. The continuity condition in (b) depends on the stability result above (which is the analogue of the stability result in [14]). The proof of (c) is more involved, but follows exactly the same route as the proof of Theorem V.4.1 (d) in [14]. It uses the continuity proven in (b) to reduce the problem to proving the Markov property (via finite-valued stopping times) which 43 is then established by using the uniqueness of solutions of the strong equation. Again, the only additional detail to worry about is the interpretation of the white noise integral J. We now prove the compact support property for solutions of (SE)Z0,K . Definition. Let f : [0,∞)× Ω̂→ (E,‖ · ‖) (a normed linear space). A bounded (Ft)-stopping time T is called a reducing time for f if and only if 1(0 < t ≤ T )‖ f (t,ω,y)‖ is uniformly bounded. We say {Tn} reduces f if and only if each Tn reduces f and Tn ↑ ∞ P-a.s. If such a sequence exists, we say f is locally bounded. Theorem 2.5.3. Assume that f ∈D1(n,d), g∈D2(m,n,d) and b : [0,∞)×Ω̂→Rn is (F̂∗t )-predictable, and all three are locally bounded. Let Z(t) = Z0 + I( f , t)+ J(g, t) +V (b, t) where V (b, t) = ∫ t 0 b(s)ds. Define K̂(·) = Kt({y : Zt(ω,y) ∈ ·}). Then for a.a. ω for each k ∈ N there is a compact set Sk(ω) ⊂ C such that supp(K̂t) ⊂ Sk ∀t ∈ [k−1,k], and furthermore, supp(K̂t) ⊂ Sk ∀t ∈ [0,k] on {ω : supp(K̂0) is compact}. The proof of this theorem is given by a small modification of the proofs of Corollaries 3.3 and 3.4 of [13] Corollary 2.5.4. Assume (Lip). Let (X ,Z) denote the solution of (SE)Z0,K . Then for a.a. ω for each k ∈ N there is a compact set S′k(ω)⊂ Rd such that supp(Xt)⊂ S′k ∀t ∈ [k−1,k], and on {ω : supp(X0) is compact}, supp(Xt)⊂ S′k ∀t ∈ [0,k]. Proof. Note that under (Lip), the coefficients σ1,σ2 and b are locally bounded and that in the above, Xt(A) = ∫ 1(Z(t) ∈ A)Kt(dy) = K̂t(pi−1t A) where pi−1t A = {y : Zt(ω,y) ∈ A} for A bounded measurable. Apply Theorem 2.5.3 and use the fact that pit : C→ Rd is a continuous mapping. Let Sk be as in Theorem 2.5.3. Then for t ∈ [k−1,k], Xt((pitSk)c) = K̂t(pi−1t ((pitSk) c)) ≤ K̂t(Sck) = 0. Hence setting S′k = pikSk gives the result in this case. Proceed similarly for the extension for {ω : S(X0) is compact}. 44 Chapter 3 A super Ornstein-Uhlenbeck process interacting with its center of mass 3.1 Introduction The existence and uniqueness of a self-interacting measure-valued diffusion that is either attracted or repelled from its centre of mass is shown below. It seems natural to consider a super Ornstein-Uhlenbeck (SOU) process with attractor (repeller) given by the centre of mass of the process initially as it is the simplest diffusion of this type. This type of model first appeared in a recent paper of Engländer [3] where a d- dimensional binary Brownian motion, with each parent giving birth to exactly two offspring and branching occurring at integral times, is used to construct a binary branching Ornstein-Uhlenbeck process where each particle is attracted (repelled) by the center of mass (COM). This is done by solving the appropriate SDE along each branch of the particle system and then stitching these solutions together. This model can be generalized such that the underlying process is a branching Brownian motion (BBM), T, (i.e. with a general offspring distribution). We might 45 then solve an SDE on each branch of T: Y ni (t) = Y n−1 p(i) (n−1)+ γ ∫ t n−1 Ȳn(s)−Y ni (s)ds+ ∫ t n−1 dBni (s), (3.1) for n− 1 < t ≤ n, where Bni labels the ith particle of T alive from time n− 1 to n, p(i) is the parent of i and Ȳn(s) = 1 τn τn ∑ i=1 Y ni (s) is the center of mass. Here τn is the population of particles alive from time n−1 to n. This constructs a branching OU system with attraction to the COM when γ > 0 and repulsion when γ < 0. It seems reasonable then, to take a scaling limit of branching particle systems of this form and expect it to converge in distribution to a measure-valued process where the representative particles behave like an OU process attracting to (repelling from) the COM of the process. Though viable, this approach will be avoided in lieu of a second method utilizing the historical stochastic calculus of Perkins [14] which is more convenient for both constructing the SOU process interacting with its COM and for proving various properties. The idea is to use a supercritical historical Brownian motion to construct the interactive SOU process by solving a certain stochastic equation. This approach for constructing interacting measure- valued diffusions was pioneered in [13] and utilized in, for example, [8]. A supercritical historical Brownian motion, K, is a stochastic process taking values in the space of measures over the space of paths in Rd . One can think of K as a supercritical superprocess which has a path-valued Brownian motion as the underlying process. That is, if Bt is a d-dimensional Brownian motion, then B̂t = B·∧t is the underlying process of K. More information about K is provided in Section 3.2. It can be shown that if a path y : [0,∞)→Rd is chosen according to Kt (loosely speaking – this is made rigorous below), then y(s) is a Brownian motion stopped at t. Then projecting down gives XKt (·) = ∫ 1(yt ∈ ·)Kt(dy), 46 a (supercritical) super Brownian motion. One can sensibly define the Ornstein-Uhlenbeck SDE driven by y according to Kt : dZs =−γZsds+dys where γ > 0 implies Zt is attracted to the origin, and γ < 0 implies repulsion. Defining Xt(·) = ∫ 1(Zt(y) ∈ ·)Kt(dy) will give an ordinary super Ornstein-Uhlenbeck process. The intuition here is that K keeps track of the underlying branching structure, and Zt is a function transform- ing a typical Brownian path into a typical Ornstein-Uhlenbeck path. Similarly, we can define a function of the path y that is an OU process with attraction to (repelling from) the COM, according to Kt and the SOU process with attraction (repulsion) to COM as follows: dYs = γ(Ȳs−Ys)ds+dys X ′t (·) = ∫ 1(Yt(y) ∈ ·)Kt(dy) where the COM is defined as Ȳs = ∫ xX ′s(dx)∫ 1X ′s(dx) . The details are given in Section 3.3 and it is shown that such a process X ′ can be constructed by a correspondence with the ordinary SOU process, X . In Section 3.4 we prove various properties of the ordinary SOU process, X , and the interacting SOU process, X ′. It is shown in Theorem 3.4.1 that the COM converges in the case where X ′ goes extinct in finite time, regardless of the value of γ . Theorem 3.4.2 shows that on the set where X ′ survives indefinitely, the COM Ȳt converges in the attractive case (where γ > 0). Theorem 3.4.3 then shows that in the extinction case the mass normalized process X ′ t X ′t (1) converges a.s. to a point as t approaches the extinction time, which is a version of a result of Tribe [20]. With the convergence of the COM on the survival set, Theorem 3.4.9 gives that 47 in the attractive case, Xt (·) Xt(1) a.s.−→ P∞ (·) as t→ ∞ where P∞ (·) is the stationary distribution of the OU process, which is an extension of Proposition 20 of Engländer and Turaev [4] and Theorem 1 of Engländer and Winter [5] for the SOU process. The correspondence between X and X ′ given in Section 3.3 is used in Theorem 3.4.11 to prove in the attractive case, X ′t (·) X ′t (1) a.s.−→ PȲ∞∞ (·) as t→ ∞, where the distribution PȲ∞∞ (·) is equal to P∞ (·) shifted to the limiting value of the COM Ȳt . In final section, in Theorem 3.5.1, we prove that the COM of the SOU process repelling from its COM converges (i.e. when γ < 0), provided the repulsion is not too strong. It is shown that the corresponding result for the SOU process with repulsion from the origin fails. This is one of the most interesting distinctions between the interactive SOU process and the non-interactive SOU process. We conclude with some conjectures, as well as a comparison with what is known for the SOU process repelling from the origin. 3.2 Background We will take K to be a supercritical historical Brownian motion. Specifically, we let K be a (∆/2,β ,1)-historical superprocess (here ∆ is the d-dimensional Laplacian), where β > 0 constant, on the probability space ( Ω,F,(Ft)t≥0 ,P ) . The β in the definition corresponds to the branching bias in the offspring distribution, and the 1 to the variance of the offspring distribution. A martingale problem characterizing K is given below. For a more thorough explanation of historical Brownian motion than found here, see Section V.2 of [14]. Let C =C(R+,Rd) with C its Borel σ -field and define Ct = {yt : y ∈C} where yt = y·∧t and Ct = σ(ys,s≤ t,y ∈C). For metric spaces E and E ′, take Cc(E,E ′) and Cb(E,E ′) to be the space of compactly supported continuous functions from E to E ′ and the space of bounded 48 continuous functions from E to E ′, respectively. For a measure space (E,E), let bE denote the set of bounded E-measurable real valued functions. Notation. For a measure µ on a space E and a measurable function f : E→R, let µ( f ) = ∫ f dµ . Furthermore, if f : E→ Rn, define µ( f ) = (µ( f1), · · · ,µ( fn)) . For a point p ∈Rd , let |p| denote Euclidean norm of p, and for a bounded function f : E→ Rd , denote ‖ f‖= supx∈E ∑di=1 | fi(x)|. It turns out that Kt is supported on Ct ⊂C a.s. and typically, Kt puts mass on those paths that are “Brownian” (until time t and fixed thereafter). As K takes values in MF(C), Kt(·) will denote integration over the y variable. Let B̂t = B(·∧ t) be the path-valued process associated with B, taking values in Ct . Then for φ ∈ bC, if s ≤ t let Ps,tφ(y) = Es,y(φ(B̂t)), where the right hand side denotes expectation at time t given that until time s, B̂ follows the path y. The weak generator, Â, of B̂ is as follows. If φ :R+×C→R we say φ ∈D(Â) if and only if φ is bounded, continuous and (Ct)-predictable, and for some Âsφ(y) with the same properties as φ , φ(t, B̂)−φ(s, B̂)− ∫ t s Ârφ(B̂)dr, t ≥ s, is a (Ct)-martingale under Ps,y for all s≥ 0,y ∈Cs. If m ∈ MF ( Rd ) , we will say K satisfies the historical martingale problem, (HMP)m, if and only if K0 = m a.s. and ∀φ ∈D(Â), Mt(φ)≡ Kt(φt)−K0(φ0)− ∫ t 0 Ks(Âsφ)ds−β ∫ t 0 Ks(φs)ds (HMP)m is a continuous (Ft)-martingale with 〈M(φ)〉t = ∫ t 0 Ks(φ 2s )ds ∀t ≥ 0, a.s. Using the martingale problem (HMP)m, one can construct an orthogonal mar- tingale measure Mt(·) with the method of Walsh [22]. Denote by P, the σ -field of 49 (Ft)-predictable sets in R+×Ω. If ψ : R+×Ω×C→ R is P×C-measurable and∫ t 0 Ks(ψ2s )ds< ∞ ∀ t ≥ 0, (3.2) then there exists a continuous local martingale Mt(ψ) with quadratic variation 〈M(ψ)〉t = ∫ t 0 Ks(ψ2s )ds. If the expectation of the term in (3.2) is finite, then Mt(ψ) is an L2-martingale. Definition. Let (Ω̂, F̂, F̂t) = (Ω×C,F×C,Ft ×Ct). Let F̂∗t denote the univer- sal completion of F̂t . If T is a bounded (Ft)-stopping time, then the normalized Campbell measure associated with T is the measure P̂T on (Ω̂, F̂) given by P̂T (A×B) = P(1AKT (B))mT (1) for A ∈ F,B ∈ C. where mT (1) = P(KT (1)). We denote sample points in Ω̂ by (ω,y). Therefore, un- der P̂T , ω has law KT (1)dP ·m−1T (1) and conditional on ω , y has law KT (·)/KT (1). Notation. Henceforth we denote the super Ornstein-Uhlenbeck process by X and the interacting super Ornstein-Uhlenbeck process by X ′. The symbol X̊ will be used as a catch-all and will vary according to the setting, but will be a measure- valued process taking values in MF ( Rd ) . Occasionally the symbol K̊ will appear and will denote a historical process different from K. 3.3 Existence and preliminary results Definition. For two (F̂∗t )-measurable processes X1 and X2, we will say that X1 = X2,K-a.e. if X1(s,ω,y) = X2(s,ω,y),∀s≤ t,Kt-a.a. y for all fixed times t ≥ 0. We say that (X ,Z) is a solution to the strong equation (SE)1Z0,K if it satisfies (SE)1Z0,K (a) Zt(ω,y) = Z0(ω,y0)+ yt − y0− γ ∫ t 0 Zs(ω,y)ds K− a.e. (b) Xt(A) = ∫ 1(Zt ∈ A)Kt(dy) ∀A ∈B(Rd),∀t ≥ 0, 50 where Zt is an (F̂∗t )-predictable process and Xt is an (Ft)-predictable process. That this equation has a pathwise unique solution does not follow immediately from Theorem 4.10 of Perkins [13]. There it is shown that equations of the above type (but with more general, interactive, drift and diffusion terms in (a)) have solutions if K is a critical historical Brownian motion. Note that in the definitions of (SE)1Z0,K and (SE) 2 Y0,K given below, part (b) is unnecessary to solve the equations. It has been included to provide an easy com- parison to the strong equation of chapter V.1 of [14]. Theorem 3.3.1. There is a pathwise unique solution (X ,Z) to (SE)1Z0,K . That is, X is unique P-a.s. and Z K-a.e. unique. Furthermore, the map t → Xt is continuous and X is a β -super-critical super Ornstein-Uhlenbeck process. Proof. Note that K̊t = e−β tKt defines a (∆/2,0,e−β t)-Historical superprocess. Let P̂1T be the Campbell measure associated with K̊ (note that if T is taken to be a random stopping time, this measure differs from P̂T ). The proof of Theorem V.4.1 of [14] with minor modifications shows that (SE)1Z0,K̊ has a pathwise unique so- lution. This is because (K3) of Theorem 2.6 of [13] shows that under P̂1T , yt is a Brownian motion stopped at time T and Proposition 2.7 of the same memoir can be used to replace Proposition 2.4 and Remark 2.5 (c) of [14] for the setting where the branching variance depends on time. Once this is established, it is simple to deduce that if (X̊ ,Z) is the solution of (SE)1Z0,K̊ , and we let Xt(·)≡ eβ t X̊t(·) = ∫ 1(Zt ∈ ·)Kt(dy) then (X ,Z) is the pathwise unique solution of (SE)1Z0,K . The only thing to check this is that Zt(ω,y) = Z0(ω,y0)+ yt − y0− γ ∫ t 0 Zs(ω,y)ds K-a.e., but this follows from the fact that K̊t  Kt ,∀t. It can be shown by using by Theorem 2.14 of [13] that X satisfies the following martingale problem: For φ ∈C2b(Rd), Mt(φ)≡ Xt(φ)−X0(φ)− ∫ t 0 ∫ −γx ·∇φ(x)+ ∆ 2 φ(x)Xs(dx)ds−β ∫ t 0 Xs(φ)ds 51 is a martingale where 〈M(φ)〉t = ∫ t 0 Xs(φ 2)ds. Then by Theorem II.5.1 of [14] this implies that X is a version of a super Ornstein-Uhlenbeck process, with initial distribution given by K0 ( Z−10 (·) ) . Remark 3.3.2. (a) Under the Lipschitz assumptions of Section V.1 of [14], one can in fact uniquely solve (SE)1Z0,K where K is a supercritical his- torical Brownian motion. The proof above can be extended with minor mod- ifications. (b) The proof of Theorem 3.3.1 essentially shows that under P̂T , T fixed, the path process y : R+× Ω̂→ Rd such that (t,(ω,y)) 7→ yt is a d-dimensional Brownian motion stopped at T . (c) Under P̂T , T fixed, Zt can be written explicitly as a function of the driving path: For t ≤ T , eγtZt = Z0+ ∫ t 0 eγsdZs+ ∫ t 0 Zs (γeγs)ds (3.3) = Z0+ ∫ t 0 eγsdys+ ∫ t 0 eγs(−γZs)ds+ ∫ t 0 Zs(γeγs)ds. Hence, Zt(y) = e−γtZ0+ ∫ t 0 e−γ(t−s)dys. We say (X ′,Y ) is the solution to the stochastic equation (SE)2Y0,K if: (SE)2Y0,K (a) Yt(ω,y) = Y0(ω,y)+ yt − y0+ γ ∫ t 0 Ȳs(ω)−Ys(ω,y)ds,K-a.a. y (b) X ′t (A) = ∫ 1(Yt ∈ A)Kt(dy),∀A ∈B(Rd),∀t ≥ 0. where Yt is an (F̂∗t )-predictable process, X ′t is an (Ft)-predictable process and Ȳt ≡ ∫ xX ′t (dx) X ′t (1) = Kt(Yt) Kt(1) , is the center of mass of X ′. Intuitively, X ′ is an interactive measure-valued diffusion where the represen- tative particles are attracted or repelled by the centre of mass of the process (de- 52 pending on the sign of γ). We will call this the SOU process with attraction (or repulsion) to the COM. It will be shown below that questions regarding (X ′,Y ) can be naturally reformulated as questions about the ordinary SOU process. Notation. Unless stated otherwise, (X ,Z) will refer to a solution of (SE)1Z0,K and (X ′,Y ) to the solution of (SE)2Y0,K . Definition. For an arbitrary F̂∗t -adapted, Rd-valued process zt , define the centre of mass (COM), z̄t , with respect to Kt as follows: z̄t = Kt(zt) Kt(1) . We also let X̃t(·)≡ Xt(·)Xt(1) = Kt(Zt ∈ ·) Kt(1) , X̃ ′t (·)≡ X ′t (·) X ′t (1) = Kt(Yt ∈ ·) Kt(1) and K̃t(·) = Kt(·)Kt(1) . Note that as K is supercritical, Kt survives indefinitely on a set of positive probability S, and goes extinct on the set of positive probability Sc. Hence we can only make sense of z̄t for t < η where η is the extinction time. Next we show that there exists a solution to (SE)2Y0,K . We cannot adapt the proof of Theorem V.4.1 of [14] to accomplish this as the Lipschitz assumptions on the drift coefficients fail to be satisfied by our model, due to the normalizing factor Kt(1) in the definition of Ȳt . Instead we will utilize the solution of (SE)1Y0,K to define a unique solution for (SE)2Y0,K . Theorem 3.3.3. There is a pathwise unique solution to (SE)2Y0,K . Proof. Suppose there exists a solution Y satisfying (SE)2Y0,K . Then under P̂T , Yt can be written as a function of the driving path y and Ȳ . Using integration by parts gives eγtYt = Y0+ ∫ t 0 eγsdYs+ ∫ t 0 γeγsYsds = Y0+ ∫ t 0 eγsdys+ ∫ t 0 γeγsȲsds 53 and hence Yt = e−γtY0+ ∫ t 0 eγ(s−t)dys+ ∫ t 0 γeγ(s−t)Ȳsds. (3.4) If (X ,Z) is the solution to (SE)1Z0,K where Z0 = Y0, then note that by Remark 3.3.2 (c), Yt = Zt + ∫ t 0 γeγ(s−t)Ȳsds. By taking the normalized measure K̃t of both sides of the above equation, we get Ȳt = Z̄t + γ ∫ t 0 e−γ(t−s)Ȳsds (3.5) Hence Ȳt is seen to satisfy an integral equation. This is a Volterra Integral Equation of the second kind (see Equation 2.2.1 of [16]) and therefore can be solved pathwise to give Ȳt = Z̄t + γ ∫ t 0 Z̄sds, which is easily verified using integration by parts. Also, if Ȳ 1t is a second process which solves (3.5), then |Ȳt − Ȳ 1t |= ∣∣∣γ ∫ t 0 e−γ(t−s)(Ȳs− Ȳ 1s )ds ∣∣∣ ≤ |γ| ∫ t 0 e−γ(t−s)|Ȳs− Ȳ 1s |ds. By Gronwall’s inequality, this implies Ȳt = Ȳ 1t , for all t andω . Pathwise uniqueness of X ′ follows from the uniqueness of the solution to (SE)1Z0,K and the uniqueness of the process Ȳt solving (3.5). We have shown that if there exists a solution to (SE)2Y0,K , then it is necessarily pathwise unique. Turning now to existence to complete the proof, we work in the opposite order and define Y and X ′ as functions of the pathwise unique solution to 54 (SE)1Z0,K where Z0 = Y0: Yt = Zt + γ ∫ t 0 Z̄sds X ′t (·) = Kt(Yt ∈ ·). Then Ȳt satisfies the integral equation (3.5), and hence∫ t 0 Z̄sds = ∫ t 0 e−γ(t−s)Ȳsds. Therefore Yt = Zt + γ ∫ t 0 e−γ(t−s)Ȳsds = e−γtY0+ ∫ t 0 e−γ(t−s)dys+ γ ∫ t 0 e−γ(t−s)Ȳsds, and so eγtYt = Y0+ ∫ t 0 eγsdys+ γ ∫ t 0 eγsȲsds. Multiplying by e−γt and using integration by parts shows Yt = Y0+ yt − y0+ γ ∫ t 0 (Ȳs−Ys)ds which holds for K-a.e. y, thereby showing (X ′,Y ) satisfies (SE)2Y0,K . Remark 3.3.4. Some useful equivalences in the above proof are collected below. If Y0 = Z0, then for t < η , (a) Yt = Zt + γ ∫ t 0 Z̄sds (b) Ȳt = Z̄t + γ ∫ t 0 Z̄sds (c) Yt − Ȳt = Zt − Z̄t (d) ∫ t 0 e−γ(t−s)Ȳsds = ∫ t 0 Z̄sds. 55 The significance of these equations is that they intimately tie the behaviour of the SOU process with attraction to its center of mass to the SOU process with attraction to the origin. Part (a) says that the SOU process with attraction to the center of mass is the same as the ordinary SOU process pushed by the position of its center of mass. Definition. Let the process Mt be defined for t ∈ [0,ζ ) where ζ ≤ ∞ possibly random. M is called a local martingale on its lifetime if there exist stopping times TN ↑ ζ such that MTN∧· is a martingale for all N. The interval [0,ζ ) is called the lifetime of M. We now consider the martingale problem for X ′. For φ : Rd → R, let φ̄t ≡ Kt(φ(Yt))/Kt(1). Note that the lifetime of the process φ̄ is [0,η). Then the follow- ing theorem holds: Theorem 3.3.5. For φ ∈C2b(Rd ,R), and t < η , φ̄t = φ̄0+Mt + ∫ t 0 b̄sds where bs = γ∇φ(Ys) · (Ȳs−Ys)+ 12∆φ(Ys) and Mt is a continuous local martingale on its lifetime such that Mt = ∫ t 0 ∫ φ(Ys)− φ̄s Ks(1) dM(s,y) and hence has quadratic variation given by [M]t = ∫ t 0 φ 2s − (φ̄s)2 Ks(1) ds. Proof. The proof is not very difficult; one need only use Itô’s Lemma followed by some slight modifications of theorems in Chapter V of [14] to deal with the drift introduced in the historical martingale problem due to the supercritical branching. Let T be a fixed time and t ≤ T . Recall that under P̂T , y is a stopped Brownian motion by Remark 3.3.2 (b), and hence Yt(y) is a stopped Ornstein-Uhlenbeck 56 process (attracting to Ȳt). Hence, by the classical Itô’s Lemma we have, under P̂T , φ(Yt) = φ(Y0)+ ∫ t 0 ∇φ(Ys) ·dYs+ 12 ∑i, j≤d ∫ t 0 φi j(Ys)d[Y i,Y j]s = φ(Y0)+ ∫ t 0 ∇φ(Ys) ·dys+ ∫ t 0 γ∇φ(Ys) · (Ȳs−Ys)+ 12∆φ(Ys)ds. Then Kt(φ(Yt)) = Kt(φ(Y0))+Kt (∫ t 0 ∇φ(Ys) ·dys ) +Kt (∫ t 0 bsds ) = K0(φ(Y0))+ ∫ t 0 ∫ φ(Y0)dM(s,y)+β ∫ t 0 Ks (φ(Y0))ds + ∫ t 0 ∫ [∫ s 0 ∇φ(Yr) ·dyr ] dM(s,y)+β ∫ t 0 Ks [∫ s 0 ∇φ(Yr) ·dyr ] ds + ∫ t 0 ∫ bsdM(s,y)+β ∫ t 0 Ks(bs)ds+ ∫ t 0 Ks(bs)ds = K0(φ(Y0))+ ∫ t 0 ∫ φ(Ys)dM(s,y) +β ∫ t 0 Ks (φ(Ys))ds+ ∫ t 0 Ks(bs)ds. The equality in the third line to Kt (∫ t 0∇φ(Ys) ·dys ) follows from Proposition 2.13 of [13]. The equality of the fourth line to the last term in the first line follows from a generalization of Proposition V.2.4 (b) of [14]. The last equality then follows from collecting like terms and using the definition of Ȳ . Supposing t <η , Itô’s formula implies (with properties of Kt(1) and Kt(φ(Yt))) φ̄t = Kt (φ(Yt)) Kt(1) = φ̄0+ ∫ t 0 ∫ [φ(Ys) Ks(1) − Ks(φ(Ys)) Ks(1)2 ] dM(s,y)+ ∫ t 0 Ks(bs) Ks(1) ds Since φ is bounded, the stochastic integral term can be localized using the stopping times TN ≡min{t : Kt(1)≥ N or Kt(1)≤ 1/N}∧N and hence it is a local martin- gale on [0,η). It is easy to check that it has the appropriate quadratic variation. Remark 3.3.6. The method of Theorem 3.3.5 can be used to show that for the 57 β -supercritical SOU process, X, for φ ∈C2b(Rd ,R) and t < η , X̃t(φ) = K̃t(φ(Zt)) = K̃0(φ(Z0))+Nt + ∫ t 0 K̃s(Lφ(Zs))ds where Lφ(x) =−γx ·∇φ(x)+ 1 2 ∆φ(x) and Nt is a continuous local martingale on its lifetime such that Nt = ∫ t 0 ∫ φ(Zs)− K̃s(φ(Zs)) Ks(1) dM(s,y) and hence has quadratic variation given by [N]t = ∫ t 0 K̃s ( φ 2(Zs) )− K̃s(φ(Zs))2 Ks(1) ds. Lemma 3.3.7. Let X be a (∆/2,β ,1) superprocess with β > 0 constant. Then there is a non-negative random variable W such that e−β tXt(1)→W a.s. and {η < ∞}= {W = 0} almost surely. Proof. Note that X̊t = e−β tXt is a (∆/2,0,e−β t)-superprocess. The martingale problem then shows that X̊t(1) is a non-negative (Ft)-martingale and therefore con- verges almost surely by the Martingale Convergence Theorem to a random variable W . It follows that {η <∞} ⊂ {W = 0}, since 0 is an absorbing state for Xt(1). Ex- ercise II.5.3 in [14] shows that P(η < ∞) = e−2βX0(1). The same exercise also shows P0,X0(exp(−λ X̊t(1))) = exp ( − 2λβ X̊0(1) 2β +λ ( 1− e−β t) ) . 58 Now sending t→ ∞ gives P0,X0(e −λW ) = exp ( −2βλ X̊0(1) 2β +λ ) , and sending λ → ∞ gives P(W = 0) = e−2β X̊0(1) = P(η < ∞) since X0 = X̊0. Therefore {η < ∞}= {W = 0} almost surely. Define h(δ ) = (δ ln+(1/δ ))1/2 where ln+(x) = lnx∨1. Let S(δ ,c) = {y : |yt− ys|< ch(|t− s|),∀t,s with |t− s| ≤ δ}. Lemma 3.3.8. Let K be a supercritical historical Brownian motion, with drift β , branching variance 1, and initial measure X0. For c0 > 6 fixed, c(t) = √ t+ c0, there exists a.s. δ (ω)> 0 such that supp(Kt(ω))⊂ S(δ (ω),c(t)) for all t. Further, given c0, P(δ < λ )< pc0(λ ) where pc0(λ ) ↓ 0 as λ ↓ 0 and for any α > 0, c0 can be chosen large enough so that pc0(λ ) =C(d,c0)λα for λ ∈ [0,1]. Proof. We follow the proof of Perkins [14], Theorem III.1.3 (a). First note that if H is another supercritical historical Brownian motion starting at time τ with initial measure m under Qτ,m and A a Borel subset of C(Rn), then the process defined by H ′t (·) = Ht (·∩{y : yτ ∈ A}) is also a supercritical historical Brownian motion starting at time τ with initial measure m′ given by m′(·) = m(·∩A) under Qτ,m. Then using the extinction prob- abilities for H ′, we have Qτ,m (Ht ({y : yτ ∈ A}) = 0 ∀t ≥ s) = exp { − 2βm(A) 1− e−β (s−τ) } . (3.6) 59 Using the Markov property for K at time j2n and (3.6) gives P [ ∃t > j+1 2n s.t. Kt ({ y : ∣∣∣∣y( j2n ) − y ( j−1 2n )∣∣∣∣> c( j2n ) h ( 2−n )}) > 0 ] (3.7) = P 1− exp −2βK j2n ({ y : ∣∣∣y( j2n)− y( j−12n )∣∣∣> c( j2n)h(2−n)}) 1− e−β 12n   ≤ P 2βK j2n ({ y : ∣∣∣y( j2n)− y( j−12n )∣∣∣> c( j2n)h(2−n)}) 1− e−β 12n  ≤ P 2βK j2n ({ y : ∣∣∣y( j2n)− y( j−12n )∣∣∣> c( j2n)h(2−n)}) β 2n − β 2 22n+1  ≤ 2 n+1 1− β2n+1 P ( K j 2n (1) ) P̂ j 2n (∣∣∣∣y( j2n ) − y ( j−1 2n )∣∣∣∣> c( j2n ) h ( 2−n )) . Since under the normalized mean measure, y is a stopped Brownian motion by Remark 3.3.2(b), we use tail estimates to get the following bound: (3.7)≤ 2 n+1 1− β2n+1 P ( K j 2n (1) ) cdnd/2−12−nc( j 2n ) 2 /2 ≤ 2n+22 β j ln22n P(X0(1))cdnd/2−12−n( j 2n +c0)/2. 60 Hence, summing over j from 1 to n2n gives P [ There exists 1≤ j ≤ n2n s.t. ∃t > j+1 2n s.t. Kt ({ y : ∣∣∣∣y( j2n ) − y ( j−1 2n )∣∣∣∣> c( j2n ) h ( 2−n )}) > 0 ] ≤ P(X0(1))cdnd/2−12n+2−nc0/2 n2n ∑ j=1 2 β j ln2 2n −n( j2n ) ≤ P(X0(1))cdnd/2−12−2n+2 n2n ∑ j=1 2− j 2n ≤ P(X0(1))cdnd/2−12−n+2, where we have used the fact that c0 > 6. Hence the sum over n of the above shows by Borel-Cantelli that there exists for almost sure ω , N(ω) such that for all n> N, for all 1≤ j ≤ n2n, for all t ≥ j+12n , for Kt-a.a. y,∣∣∣∣y( j−12n ) − y ( j 2n )∣∣∣∣< c( j2n ) h ( 2−n ) . Letting δ (ω) = 2−N(ω), note that on the dyadics, by above, we have that P(δ < λ ) = P ( N >− lnλ ln2 ) = P [ ∃n>− lnλ ln2 ,∃ j ≤ n2n,∃t > j+1 2n , s.t. Kt ({ y : ∣∣∣∣y( j2n ) − y ( j−1 2n )∣∣∣∣> c( j2n ) h ( 2−n )}) > 0 ] ≤ ∑ n=b− lnλln2 c C′(d,c0)nd/2−122n−nc0/2 ≤C(d,c0,ε)λ c0 2 −ε where ε can be chosen to be arbitrarily small (though the constant C will increase as it decreases). The rest of the proof follows as in Theorem III.1.3(a) of [14], via an argument similar to Levy’s proof for the modulus of continuity for Brownian motion. 61 The following moment estimates are useful in establishing the convergence of Ȳt . Recall that η is the extinction time of K. Lemma 3.3.9. Assume P ( K̃0 (|Z0|2+ |y0|2))< ∞. Then, P ( |Yt |2; t < η ) < A(γ, t), where A(γ, t) =  O ( 1+ t6e−2γt ) if γ < 0 O ( 1+ t5 ) if γ ≥ 0. Proof. Assume that t < η . Recall Zt(ω,y)≡ e−γtZ0+ ∫ t 0 eγ(s−t)dys. Note that below ‘.’ denotes less than up to multiplicative constants independent of t and y. Suppose that y ∈ S(δ ,c(t)), where S(δ ,c(t)) is the same as in the previous lemma. Then, as Yt = Zt + γ ∫ t 0 Z̄sds, |Yt |2 . |Zt |2+ γ2t ∫ t 0 |Z̄s|2 ds . |Zt |2+ γ2t ∫ t 0 |Zs|2ds by Cauchy-Schwartz and Jensen’s inequality. Therefore integrating with respect to the normalized measure gives |Yt |2 ≤ |Zt |2+ γ2t ∫ t 0 |Zs|2ds (3.8) and therefore we need only find the appropriate bounds for expectation of |Zt |2 to get the result. 62 After another few applications of Cauchy-Schwartz and integrating by parts, |Zt |2 . e−2γt |Z0|2+ ∣∣∣e−γt ∫ t 0 eγsdys ∣∣∣2 . e−2γt |Z0|2+ e−2γt ∣∣∣eγtyt − y0− γ ∫ t 0 yseγsds ∣∣∣2 . e−2γt (|Z0|2+ |y0|2)+ |yt |2+ γ2t ∫ t 0 |ys|2e−2γ(t−s)ds. As y ∈ S(δ ,c(t)), |Zt |2 . e−2γt (|Z0|2+ |y0|2)+ |y0|2+( tc(t)h(δ )δ )2 + γ2t ∫ t 0 [ |y0|2+ ( sc(s)h(δ ) δ )2] e−2γ(t−s)ds . e−2γt (|Z0|2+ |y0|2)+ |y0|2+ |y0|2γt (1− e−2γt)/2 + c(t)2 ( h(δ ) δ )2 ( t2+ γt3 ( 1− e−2γt)) . (1+ |γ| t)(1+ e−2γt)(|Z0|2+ |y0|2) + c(t)2 ( h(δ ) δ )2 ( t2+ γt3 ( 1− e−2γt)) Integrating by the normalized measure K̃t , |Zt |2 . (1+ |γ| t) ( 1+ e−2γt ) K̃t (|Z0|2+ |y0|2) + c(t)2 ( h(δ ) δ )2 ( t2+ γt3 ( 1− e−2γt)) . Then using (3.8) and using the above bound on |Zt |2 gives |Yt |2 ≤ |Zt |2+ γ2t ∫ t 0 |Zs|2ds . (1+ t2) ( 1+ e−2γt )( K̃t (|Z0|2+ |y0|2)+∫ t 0 K̃s (|Z0|2+ |y0|2)ds) (3.9) +(1+ t)c(t)2 ( h(δ ) δ )2 ( t2+ γt3 ( 1− e−2γt)) . 63 Note that φ(y) ≡ |Z0(y)|2 + |y0|2 and φn(y) ≡ φ(y)1(|y| ≤ n) are F̂0 measurable. By applying Itô’s formula to Kt(φn)K−1t (1) and using the decomposition Kt(φn) = K0(φn)+ ∫ t 0 φn(y)dM(s,y)+β ∫ t 0 Ks(φn)ds (which follows from Proposition 2.7 of [13]) we get K̃t(φn) = K̃0(φn)+Nt(φn), where Nt(φn) is a local martingale until time η , for each n. In fact the sequence of stopping times {TN} appearing in Theorem 3.3.5 can be used to localize each Nt(φn). Applying first the Monotone Convergence Theorem and then localizing gives, P ( K̃t(φ); t < η ) = lim n→∞P ( K̃t(φn); t < η ) = lim n→∞ limN→∞ P ( K̃t(φn)1(t < TN) ) = lim n→∞ limN→∞ P ( K̃t∧TN (φn)− K̃TN (φn)1(t ≥ TN) ) ≤ lim n→∞ limN→∞ P ( K̃t∧TN (φn) ) = lim n→∞ limN→∞ P ( K̃0(φn)+Nt∧TN (φn) ) = lim n→∞P ( K̃0(φn) ) = P ( K̃0(φ) ) , where we have used the positivity of φn to get the fourth line and the Monotone Convergence Theorem in the last line. Further, note that P (∫ t 0 K̃s(φ)ds; t < η ) ≤ P (∫ t 0 K̃s(φ)1(s< η)ds ) = ∫ t 0 P ( K̃s(φ);s< η ) ds ≤ tP(K̃0(φ)) , by the calculation immediately above. Therefore, taking expectations in (3.9) and 64 plugging in c(t) = √ c0+ t gives P ( |Yt |2; t < η ) . ( 1+ t3 )( 1+ e−2γt ) P ( K̃0 (φ) ) + t5 ( 1+ te−2γt ) P ( h(δ )2 δ 2 ) . Now let c0 be chosen so that supp(Kt)⊂ S(δ ,c(t)) and pc0(λ ) =Cλα for λ ∈ [0,1]. Note that P ( h(δ )2 δ 2 ) = P ( ln+(1/δ ) δ ) = ∫ ∞ 0 ( ln+(1/λ ) λ ) dP(δ < λ ) ≤ ∫ 1 0 ( ln+(1/λ ) λ ) dP(δ < λ )+P(δ > 1) = lim λ↓0 ( ln+(1/λ ) λ P(δ < λ ) ) −P(δ < 1) − ∫ 1 0 P(δ < λ )d ( ln+(1/λ ) λ ) +P(δ > 1) < ∞, by choosing the original constant c0 so that α is large (α ≥ 2 is enough). Remark 3.3.10. (a) The proof of Lemma 3.3.9, under the same hypotheses yields P ( |Zt |2; t < η ) < A(γ, t). (b) Lemma 3.3.9 and its proof can be extended to show that for any positive in- teger k if P ( K̃0 (|Z0|k + |y0|k)) < ∞, then there exists a function B(γ, t,k) polynomial in t if γ ≥ 0, exponential if γ < 0 such that P ( |Yt |k; t < η ) < B(γ, t,k) and P ( |Zt |k; t < η ) < B(γ, t,k). We only need to note that the exponent α in Lemma 3.3.8 can be made arbi- trarily large by choosing a sufficiently large constant c0. Hence by choosing 65 α appropriately, we can show that P [( h(δ ) δ )k] < ∞, which can then be used to adapt the proof above. 3.4 Convergence For this section we will assume that Z0(y) = Y0(y) = y0. Then by the definition of X and X ′, ∫ φ(y0)K0(dy) = ∫ φ(x)X0(dx) = ∫ φ(x)X ′0(dx). Assume that X ′0(1) = X0(1)< ∞ for Theorems 3.4.1, 3.4.2 and 3.4.3 below. Theorem 3.4.1. On Sc, Ȳt and Z̄t converge as t ↑ η < ∞, P-a.s., for any γ ∈ R. Proof. Assume for now that P ( K̃0 ( |y0|2 )) < ∞ (the case where K0 = 0 can be ignored without loss of generality). By Theorem 3.3.5, Ȳt = Ȳ0+ ∫ t 0 ∫ Ys− Ȳs Ks(1) dM(s,y) and therefore is a local martingale on its lifetime with reducing sequence {TN} as defined in the proof of the same theorem. Using Doob’s weak inequality and Lemma 3.3.9, P ( sup s<t∧η |Ȳs|> n ) = lim N→∞ P ( sup s<t∧TN |Ȳs|> n ) ≤ lim N→∞ 1 n2 E(|Ȳt∧TN |2) < A(γ, t) n2 . By the first Borel-Cantelli lemma, P ( sup s<t∧η |Ȳt |> n i.o. ) = 0. It follows that liminf s→t∧η Ȳs >−∞ and limsups→t∧η Ȳs < ∞ 66 which implies that on the set {η < t}, Ȳs converges, by Theorem IV.34.12 of Rogers and Williams [18]. This shows convergence on the extinction set as Sc =∪t{η < t}. Note that if ν(·) = P(K0 ∈ ·). Theorem II.8.3 of [14] gives P(K ∈ ·) = ∫ PK0 (K ∈ ·)dν(K0), where PK0 is the law of a historical Brownian motion with initial distribution δK0 . Hence, the a.s. convergence of Ȳt in the case where K̃0 (|y0|2) is finite in mean imply P ( lim t↑η Ȳt exists;Sc ) = ∫ PK0 ( lim t↑η Ȳt exists;Sc ) dν(K0) (3.10) = ∫ PK0 (S c)dν(K0) = P(Sc) if K0 (|y0|2+1)< ∞,ν-a.s. Finally, to get rid of the assumption that K0 (|y0|2)< ∞ note that Corollary 3.4 of [13] ensures that if K0(1) < ∞, then at any time t > 0, Kt (and hence Xt ,X ′t ) is compactly supported. Therefore, letting Sr = {Kr 6= 0} we see that P ( lim t↑η Ȳt does not exist,Sc ) = P (⋃ r∈N { lim t↑η Ȳt does not exist,S1/r,S c }) ≤ ∑ r∈N P ( PK1/r ( lim t↑η Ȳt does not exist,Sc ) 1 ( S1/r )) = 0 by 3.10 since K1/r a.s. compact implies that K1/r (|y1/r|2) < ∞ holds. This com- pletes the proof for the convergence of Ȳ on Sc in its full generality. The convergence of Z̄t now follows from the convergence of Ȳt and Equa- tion 3.5. Theorem 3.4.2. On S the following hold: 67 (a) If γ > 0 Z̄t a.s.−→ 0 and Ȳt a.s.−→ γ ∫ ∞ 0 Z̄sds, and this integral is finite almost surely. (b) If γ = 0, then Z̄t = Ȳt converges almost surely. Proof. Let γ ≥ 0 and as in the previous theorem assume P ( K̃0 ( |y0|2 )) <∞. Also, without loss of generality, assume that d = 1 for this proof. By Theorem 3.3.5, Ȳt is a continuous local martingale with decomposition given by Ȳt = Ȳ0+Mt(Y ) where [M(Y )]t = ∫ t 0 Y 2s − Ȳ 2s Ks(1) ds≡ ∫ t 0 V (Ys) Ks(1) ds. Theorem IV.34.12 of [18] shows that on the set {[M(Y )]∞ < ∞} ∩ S, Mt(Y ) a.s. converges. Note that by Lemma 3.3.7, for a.s. ω ∈ S, W (ω) > 0, recalling that W = limt→∞ e−β tKt(1). Hence it follows that [M(Y )]∞ < ∞ on S if∫ ∞ 0 e−β sV (Ys)ds< ∞. Then P (∫ ∞ 0 V (Ys) eβ s ds;S ) ≤ P (∫ ∞ 0 Y 2s eβ s ds;S ) ≤ ∫ ∞ 0 A(γ,s) eβ s ds < ∞ by Cauchy-Schwartz and Lemma 3.3.9 since γ ≥ 0. Therefore on S, Ȳt converges a.s. to some limit Ȳ∞. Note that if γ = 0, Remark 3.3.4 (b) gives Ȳt = Z̄t and so (b) holds. That Z̄t converges on S for γ > 0 follows from the fact that Ȳt converges and 68 Equation 3.5 by setting Z̄t = Ȳt − γ ∫ t 0 e−γ(t−s)Ȳsds = Ȳt − Ȳ∞+ γ ∫ t 0 e−γ(t−s) (Ȳ∞− Ȳs)ds+ e−γtȲ∞ → 0 as t→ ∞. By Remark 3.3.4 (b) we see that for γ > 0 since Ȳt = Z̄t + γ ∫ t 0 Z̄sds, Z̄t a.s.−→ 0 and Ȳt a.s.−→ γ ∫ ∞0 Z̄sds. Now argue by conditioning as in the end of Theorem 3.4.1 to get the full result. Theorem 3.4.3. On the extinction set, Sc, X̃t → δF and X̃ ′t → δF ′ as t ↑ η < ∞ a.s., where F and F ′ are Rd-valued random variables such that F ′ = F + γ ∫ η 0 Z̄sds. Proof. As in the previous theorems, note that we need only consider the case that P ( K̃0 ( |y0|2 )) < ∞. We will follow the proof of Theorem 1 of Tribe [20] here. Define ζ (t) = ∫ t 0 1 Ks(1) ds, t < η . It is known by the work of Konno and Shiga [10] in the case where β = 0, that ζ : [0,η)→ [0,∞) homeomorphically (recall that η < ∞ a.s. in that case). This latter result also holds when β > 0 on the extinction set, Sc, by a Girsanov argument. To see this, suppose that on some probability space, zt = z0+ ∫ t 0 √ zsdBs,Q-a.s. Then using Girsanov’s Theorem, (Theorem V.27.1 of [18]) we see that there is a 69 probability measure Q′ such that, dQdQ′ |Ft = Nt where Nt = exp (∫ t 0 βdzs− β 2 2 ∫ t 0 zsds ) is an exponential martingale. Furthermore, under Q′, zt solves zt = z0+ ∫ t 0 √ zsdBs+ ∫ t 0 β zsds,Q′-a.s. Let Sct = {ω : ∃s≤ t,zs = 0}. Then∫ Sct 1 (∫ η 0 1 zs ds< ∞ ) dQ′ = ∫ Sct 1 (∫ η 0 1 zs ds< ∞ ) NtdQ= 0, by the result of Konno and Shiga. Note then that on ∪Sct , ∫ η 0 z −1 s ds = ∞, Q′-a.s. Since Kt(1) is a diffusion that satisfies the previous SDE with respect to P, ζ satis- fies the required properties on Sc, P-a.s. as well as on S (where it is trivial). Define D : [0,ζ (η−))→ [0,η) as the unique inverse of ζ (on Sc, this defines the inverse on [0,∞)) and for t ≥ ζ (η−), let Dt = ∞. Let XDt = X ′ Dt , X̃ D t = XDt XDt (1) and Gt = FDt and define Ltφ(x) = γ(Ȳt − x) ·∇φ(x)+ 12∆φ(x). Let TN = ∫ ηN 0 1 Ks(1) ds, where ηN = inf{s : Ks(1)≤ 1/N}. Then note that TN ↑ ζ (η−) and each TN is a 70 Gt-stopping time. On Sc for φ ∈C2b , Theorem 3.3.5 implies, X̃Dt∧TN (φ) = X̃0(φ)+ ∫ Dt∧TN 0 X̃ ′s(Lsφ)ds+MDt∧TN (φ) = X̃0(φ)+ ∫ t∧TN 0 X̃Ds (LDsφ)X D s (1)ds+Nt∧TN (φ) = X̃0(φ)+ ∫ t∧TN 0 XDs (LDsφ)ds+Nt∧TN (φ) (3.11) since dDt = XDt (1)dt and where Nt = MDt . It follows that Nt∧TN is a Gt-local mar- tingale. Then, by Theorem 3.3.5, [ X̃D(φ) ] t∧TN = ∫ Dt∧TN 0 X̃ ′s(φ 2)− X̃ ′s(φ)2 Xs(1) ds = ∫ t∧TN 0 X̃Ds (φ 2)− X̃Ds (φ)2ds, which is uniformly bounded in N. Hence, sending N→ ∞, one sees that Nt∧ζ (η−) is a Gt-martingale. Note that on Sc, ζ (η−) = ∞ and hence on that event,∫ ∞ 0 XDs (|LDsφ |)ds≤ ∫ ∞ 0 ∫ |γ(ȲDs− x) ·∇φ(x)|+ 1 2 |∆φ(x)|XDs (dx)ds ≤ ∫ ∞ 0 |γ|‖∇φ‖KDs(|ȲDs−YDs |)+ 1 2 ‖∆φ‖XDs (1)ds = ∫ ∞ 0 XDs (1) ( |γ|‖∇φ‖K̃Ds (|ȲDs−YDs |)+ 1 2 ‖∆φ‖ ) ds ≤ ∫ ∞ 0 XDs (1) ( |γ|‖∇φ‖ ( |YDs |2 ) 1 2 + 1 2 ‖∆φ‖ ) ds where in the second line we have used the definition of X ′ and the Cauchy-Schwartz 71 inequality in the fourth. Using the definition of Ds yields∫ ∞ 0 XDs (|LDsφ |)ds≤ ( |γ|‖∇φ‖sup s<η ( |Ys|2 ) 1 2 + 1 2 ‖∆φ‖ )∫ ∞ 0 XDs (1)ds = ( |γ|‖∇φ‖sup s<η ( |Ys|2 ) 1 2 + 1 2 ‖∆φ‖ ) η < ∞ (3.12) as φ ∈ C2b and |Ys|2 is continuous on [0,η) (which follows from Theorem 3.3.5). Hence, this implies that for φ positive, on Sc Nt(φ)>−X̃0(φ)− ∫ ∞ 0 XDs (|LDsφ |)ds for all t and hence by Corollary IV.34.13 of [18], Nt converges as t→∞. Therefore by (3.11) and (3.12), X̃Dt (φ) converges a.s. as well. Denote by X̃D∞ (φ) the limit of X̃Dt (φ). It is immediately evident that X̃D∞ (·) is a probability measure on Rd . To show that X̃D∞ (·) = δF ′ where F ′ is a random point in Rd , we now defer to the proof of Theorem 1 in Tribe [20], as it is identical from this point forwards. Similar (but simpler) reasoning holds to show X̃t → δF a.s. on Sc where F is a random point in Rd . Let f (t) = γ ∫ t 0 Z̄sds. Note that f is independent of y and that f (t)→ f (η) a.s. when t ↑ η because Ȳt = Z̄t + f (t) and both Ȳt and Z̄t converge a.s. by Theorem 3.4.1. Then for φ bounded and Lipschitz,∣∣∣∣∫ φ(x− f (t))X̃ ′t (dx)−∫ φ(x− f (η))X̃ ′t (dx)∣∣∣∣≤C | f (η)− f (t)| a.s.−→ 0. as t ↑ η . Therefore it is enough to note that since f (η) depends only on ω , the convergence of X̃ ′t gives∫ φ(x− f (η))X̃ ′t (dx) a.s.−→ φ ( F ′− f (η)) 72 and hence ∫ φ(x− f (t))X̃ ′t (dx) a.s.−→ φ ( F ′− f (η)) . By Remark 3.3.4 (a),∫ φ(x− f (t))X̃ ′t (dx) = ∫ φ (Yt(y)− f (t)) K̃t(dy) = ∫ φ (Zt(y)) K̃t(dy) = ∫ φ(x)X̃t(dx) a.s.−→ φ (F) as t ↑ η . Since there exists a countable separating set of bounded Lipschitz func- tions {φn}, and the above holds for each φn, F ′ = F + γ ∫ η 0 Z̄sds, a.s. Remark 3.4.4. (a) Theorem 3.4.3 holds in the critical branching case. That is, if β = 0, Xt a.s.−→ δF and X ′t a.s.−→ δF ′ where F ′ = F + ∫ η 0 Z̄sds. The convergence of the critical ordinary SOU process to a random point follows directly from Tribe’s result. That this holds for the SOU process with attraction to the COM follows from the calculations above. (b) The distribution of the random point F has been identified in Tribe [20] by approximating with branching particle systems. In fact, the law of F can be identified as xη , where xt is an Ornstein-Uhlenbeck process with initial distribution given by X̃0 and η is the extinction time. Finding the distribution of F ′ remains an open problem however. The next few results are necessary to establish the almost sure convergence of 73 X̃t on the survival set. This will in turn be used to show the almost sure convergence of X̃ ′t using the correspondence of Remark 3.3.4 (a). Define Lip1 = { ψ ∈C(Rd) : ∀x,y, |ψ(x)−ψ(y)| ≤ |x− y| ,‖ψ‖ ≤ 1}. Define Pt as the standard Ornstein-Uhlenbeck semigroup (with attraction to the origin). Note that Pt → P∞ in norm where P∞φ(x) = ∫ φ(z) ( γ pi ) d 2 e−γ|z| 2 dz, which is independent of x. Recall that W = limt→∞ e−β tXt(1) and S= {W > 0} a.s. from Lemma 3.3.7. For the following pair of lemmas we will, with a slight abuse of notation, allow M to denote the orthogonal martingale measure generated by the martingale prob- lem for X . Let A be the infinitesimal generator for an Ornstein-Uhlenbeck process, and hence recall that for φ ∈C2(Rd), Aφ(x) =−γx ·∇φ(x)+ ∆ 2 φ(x). Lemma 3.4.5. If γ > 0, P ( X0 ( |x|4 )) <∞ and P ( X0(1)4 ) <∞, then on S, for any φ ∈ Lip1, e−β tXt(φ) L 2−→WP∞φ and P (∣∣∣e−β tXt(φ)−WP∞φ ∣∣∣2)≤Ce−ζ t where C depends only on d and X0, and ζ is a positive constant dependent only on β and γ . Proof. Let φ ∈ Lip1. By the extension of the martingale problem for X given in Proposition II.5.7 of [14], for functions ψ : [0,T ]×Rd → R such that ψ satisfies the definition before that Proposition, Xt(ψt) = X0(ψ0)+ ∫ t 0 ∫ ψs(x)dM(x,s)+ ∫ t 0 Xs (Aψs+βψs+ ψ̇s)ds where M is the orthogonal martingale measure derived from the martingale prob- lem for the SOU process. It is not difficult to show that ψs = Pt−sφ where φ as 74 above satisfies requirements for Proposition II.5.7 of [14]. Plugging this in gives Xt(φ) = X0(Ptφ)+ ∫ t 0 ∫ Pt−sφ(x)dM(s,x)+ ∫ t 0 βXs(Pt−sφ)ds since ∂∂ s Psφ = APsφ . Multiplying by e −β t and integrating by parts gives e−β tXt(φ) = e−β tXt(ψt) = X0(ψ0)− ∫ t 0 βe−β sXs(ψs)ds+ ∫ t 0 e−β sdXs(ψs) = X0(Ptφ)+ ∫ t 0 ∫ e−β sPt−sφ(x)dM(s,x) (3.13) Note that as the OU-process has a stationary distribution P∞ where Pt → P∞ in norm. When s is large in (3.13), Pt−sφ(x) does not contribute much to the stochastic integral and hence we expect the limit of e−β tXt(φ) to be X0 (P∞φ)+ ∫ ∞ 0 ∫ e−β sP∞φ(x)dM(s,x), (3.14) which is a well defined, finite random variable as[∫ · 0 ∫ e−β sP∞φ(x)dM(s,x) ] ∞ < ‖φ‖2 ∫ ∞ 0 e−2β sXs(1)ds, which is finite in expectation. As P∞φ(x) does not depend on x, it follows that (3.14) = (P∞φ)X0(1)+(P∞φ) ∫ ∞ 0 ∫ e−β sdM(s,x) =WP∞φ . Given this decomposition for WP∞φ , we write P (( e−β tXt(φ)−WP∞φ )2) ≤ 3P ((∫ ∞ t e−β sP∞φ(x)dM(s,x) )2) +3P ((∫ t 0 ∫ e−β s (Pt−sφ(x)−P∞φ(x))dM(s,x) )2 +X0(P∞φ −Ptφ)2 ) If zt is a d-dimensional OU process satisfying dzt =−γztdt+dBt , where Bt is 75 a d-dimensional Brownian motion, then zt = e−γtz0+ ∫ t 0 e−(t−s)γdBt and hence zt is Gaussian, with mean e−γtz0 and covariance matrix 12γ (1− e−2γt)I. Evidently, z∞ is also Gaussian, mean 0 and variance 12γ I. We use a simple coupling: suppose that wt is a random variable independent of zt such that z∞ = zt +wt (i.e. wt is Gaussian with mean −e−γtz0 and covariance 12γ e−2γtI). Then using the fact that φ ∈ Lip1 and the Cauchy-Schwartz inequality, followed by our coupling with z0 = x gives X0(P∞φ −Ptφ)2 = (∫ Ex (φ(z∞)−φ(zt))X0(dx) )2 ≤ ∫ Ex (|z∞− zt |)2 X0(dx)X0(1) = ∫ Ex (|wt |2)X0(dx)X0(1) = ∫ e−2γt ( |x|2+ d 2γ ) X0(dx)X0(1) ≤ ce−2γt (∫ |x|2 X0(dx)X0(1)+X0(1)2 ) . Taking expectations and using Cauchy-Schwartz and the assumptions on X0 gives exponential rate of convergence for the above term. Since we can think of ∫ r 0 ∫ e−β sPt−sφ(x)dM(s,x) as a martingale in r up until time t, various martingale inequalities can be applied to get bounds for the termi- nal element, ∫ t 0 ∫ e−β sPt−sφ(x)dM(s,x). Note that this process is not in general a martingale in t. Therefore, we have P [(∫ t 0 ∫ e−β sP∞φ(x)dM(s,x)− ∫ t 0 ∫ e−β sPt−sφ(x)dM(s,x) )2] (3.15) = P [(∫ t 0 ∫ e−β s(P∞φ(x)−Pt−sφ(x))dM(s,x) )2] ≤ P [∫ t 0 e−2β s ∫ (P∞φ(x)−Pt−sφ(x))2 Xs(dx)ds ] . 76 Then as φ Lipschitz, by the coupling above, (3.15)≤ P [∫ t 0 e−2β s ∫ e−2γ(t−s) ( |x|2+ d 2γ ) Xs(dx)ds ] = ∫ t 0 e−2β s−2γ(t−s)P [∫ |x|2+ d 2γ Xs(dx) ] ds = ∫ t 0 e−2β s−2γ(t−s)P [ Ks (|Zs|2)+ d2γ Xs(1) ] ds. Applying the Cauchy-Schwartz inequality followed by Remark 3.3.10(b) gives P ( Ks (|Zs|2))= P(|Zs|2Ks(1);s< η) ≤ P ( |Zs|22;s< η ) 1 2 P ( Xs(1)2 ) 1 2 ≤ B(s,γ,4) 12P(Xs(1)2) 12 ≤ cB(s,γ,4) 12 eβ sP ( X0(1)2+ 1 β X0(1) ) 1 2 where the last line follows by first noting that e−β tXt(1) = X0(1)+ ∫ t 0 ∫ e−β sdM(s,x) is a martingale. That is, e−2β sP ( Xs(1)2 )≤ 2P(X0(1)2+(∫ s 0 e−β rdM(r,x) )2) (3.16) = 2P ( X0(1)2+ [∫ · 0 e−β rdM(r,x) ] s ) = 2P ( X0(1)2+ ∫ s 0 e−2β rXr(1)dr ) = 2P ( X0(1)2 ) +2 ∫ s 0 e−β rP ( e−β rXr(1) ) dr = 2P ( X0(1)2 ) +2 ∫ s 0 e−β rP(X0(1))dr ≤ 2P ( X0(1)2+ 1 β X0(1) ) . 77 Therefore, (3.15)≤ ∫ t 0 e−2β s−2γ(t−s) [ eβ sB(s,γ,4) 1 2P ( X0(1)2+ 1 β X0(1) ) 1 2 ] ds + ∫ t 0 e−2β s−2γ(t−s)eβ s d 2γ P(X0(1))ds ≤ ∫ t 0 e−β s−2γ(t−s) [ B(s,γ,4) 1 2P ( X0(1)2+ 1 β X0(1) ) 1 2 + d 2γ P(X0(1)) ] ds <Ce−ζ1t , where ζ1 = min(β ,2γ)− ε where ε is arbitrary small and comes from the polyno- mial term in the integral. Finally, P ((∫ ∞ t e−β sP∞φ(x)dM(s,x) )2) = (P∞φ)2P (∫ ∞ t e−2β sXs(1)ds ) ≤ (P∞φ)2 ∫ ∞ t e−β sP ( e−β sXs(1) ) ds ≤ (P∞φ) 2 β e−β tP(X0(1)) , since e−β sXs(1) is a martingale. Therefore, since ζ1 < β , we see that ζ = ζ1 gives the correct exponent. Remark 3.4.6. As the L2 convergence in Lemma 3.4.5 is exponentially fast, it follows from the Borel-Cantelli Lemma and Chebyshev inequality that for a strictly increasing sequence {tn}∞n=0 where ∣∣{tn}∩ [k,k+1)∣∣= beζk/2c, for φ ∈ Lip1, e−β tnXtn(φ)→WP∞φ a.s. as n→ ∞. The idea is to use the above remark to bootstrap up to almost sure convergence in Lemma 3.4.5 with some estimates on the modulus of continuity of the process e−β tXt(φ). Lemma 3.4.7. Suppose γ > 0, P ( X0 (|x|8))< ∞ and P(X0(1)8)< ∞. If φ ∈ Lip1 78 and h> 0, then P ([ e−β (t+h)Xt+h(φ)− e−β tXt(φ) ]4)≤C(t)h2e−ζ ∗t where ζ ∗ is a positive constant depending only on β and γ and C is polynomial in t, and depends on γ , β and d. Proof. The proof will follow in a manner very similar to the proof of the previous lemma. From the calculations above we see that e−β (t+h)Xt+h(φ)− e−β tXt(φ) = X0(Pt+hφ −Ptφ) + ∫ t+h 0 ∫ e−β sPt+h−sφ(x)dM(s,x)− ∫ t 0 ∫ e−β sPt−sφ(x)dM(s,x) = X0(Pt+hφ −Ptφ)+ ∫ t 0 ∫ e−β s ((Pt+h−s−Pt−s)φ(x))dM(s,x) + ∫ t+h t ∫ e−β sPt+h−sφ(x)dM(s,x) ≡ I1+ I2+ I3 where Pt is the OU semigroup. Using the Cauchy-Schwartz inequality, we can find bounds for P(|Ik|4),k = 1,2,3 separately. |I1|4 = X0(Pt+hφ −Ptφ)4 ≤ [X0 ((Pt+hφ −Ptφ)2)X0(1)]2 Recalling the simple coupling in the previous lemma to see that (Pt+hφ(x)−Ptφ(x))2 ≤ Ex (|φ(zt+h)−φ(zt)|2) ≤ Ex (|zt+h− zt |2) ≤ Ex (|wt,t+h|2) where z is as above, an OU process started at x, and ws,t is independent of zs but such that zt = zs +ws,t Hence ws,t is Gaussian with mean x(e−γt − e−γs) and 79 covariance matrix I2γ (e −2sγ − e−2tγ). Therefore, (Pt+hφ(x)−Ptφ(x))2 ≤ |x|2 ( e−γ(t+h)− e−γt )2 + d 2γ ( e−2tγ − e−2(t+h)γ ) = e−2γt ( |x|2 ( 1− e−γh )2 + d 2γ ( 1− e−2hγ )) . Hence P(|I1|4) = P [( e−2γt ( 1− e−γh )2 ∫ |x|2X0(dx)X0(1)+ d2γ e −2γt ( 1− e2γh ) X0(1)2 )2] ≤C1(d,γ)h2e−4γt where C1 is a constant dependent on X0 (·), and is finite by assumptions on the initial measure. To get bounds on the expectation of I2 we use martingale inequalities. Note that∫ · 0 ∫ e−β s ((Pt+h−s−Pt−s)φ(x))dM(s,x) = N(·) is a martingale until time t. There- fore, using the Burkholder-Davis-Gundy inequality and the coupling above gives P (|I2|4)≤ cP([N]2t ) = cP [(∫ t 0 ∫ e−2β s ((Pt+h−s−Pt−s)φ(x))2 Xs(dx)ds )2] ≤ cP [(∫ t 0 e−2β s ( e−γ(t+h−s)− e−γ(t−s) )2 ∫ |x|2Xs(dx)ds)2 + (∫ t 0 de−2β s 2γ ( e−2(t−s)γ − e−2(t+h−s)γ ) Xs(1)ds )2] ≤ cP [( t ∫ t 0 e−4β s ( e−γ(t+h−s)− e−γ(t−s) )4(∫ |x|2Xs(dx))2 ds) + t ( d 2γ )2 ∫ t 0 e−4β s ( e−2(t−s)γ − e−2(t+h−s)γ )2 Xs(1)2ds ] 80 = ct ∫ t 0 e−2β s ( e−γ(t+h−s)− e−γ(t−s) )4 P [(∫ |x|2e−β sXs(dx) )2] ds + t ( d 2γ )2 ∫ t 0 e−2β s ( e−2(t−s)γ − e−2(t+h−s)γ )2 P [( e−β sXs(1) )2] ds. Since Xs(|x|2) = Ks(|Zs|2), by Remark 3.3.10 (b), P ( e−2β sXs(|x|2)2 ) ≤ P [ Z2s 2( e−β sXs(1) )2 ;s< η ] ≤ P ( Z2s 4 ;s< η )1/2 P ( e−4β sXs(1)4 )1/2 ≤ P ( Z8s ;s< η )1/2 P ( e−4β sXs(1)4 )1/2 ≤ cB(γ,s,8)1/2 (P(X0(1)4+ sX0(1)2+ sX0(1)))1/2 . The bound on the expectation of e−4β sXs(1)4 follows by an application of the Burkholder-Davis-Gundy inequality to e−β sXs(1) = X0(1)+ ∫ s 0 e −β rdM(r,x): P (( e−β sXs(1) )4) ≤ cP ( X0(1)4+ (∫ s 0 e−β rdM(r,x) )4) ≤ cP ( X0(1)4+ [∫ · 0 e−β rdM(r,x) ]2 s ) ≤ cP ( X0(1)4+ (∫ s 0 e−2β rXr(1)dr )2) , where the last line follows as [M(φ)]t = ∫ t 0 Xs ( φ 2 ) ds. Then, using the Cauchy- 81 Schwartz Inequality followed by Fubini’s Theorem, P (( e−β sXs(1) )4) ≤ cP ( X0(1)4+ s ∫ s 0 e−4β rXr(1)2dr ) ≤ cP(X0(1)4)+ cs∫ s 0 e−2β rP ( e−2β rXr(1)2 ) dr = cP ( X0(1)4 ) + cs ∫ s 0 e−2β rP ( X0(1)2+ 1 β X0(1) ) dr ≤ cP ( X0(1)4+ s 2β X0(1)2+ s 2β 2 X0(1) ) . where in the third line we have substituted the calculation from (3.16). Therefore, P (|I2|4)≤ ct ∫ t 0 e−2β sB(γ,s,8)1/2 ( e−γ(t+h−s)− e−γ(t−s) )4 · (P(X0(1)4+ sX0(1)2+ sX0(1)))1/2 ds + ct 4γ2 ∫ t 0 e−2β s ( e−2(t−s)γ − e−2(t+h−s)γ )2 P ( X0(1)2+ 1 β X0(1) ) ds ≤ ctB(γ, t,8)1/2 (P(X0(1)4+ tX0(1)2+ tX0(1)))1/2 · ( e−γh−1 )4 ∫ t 0 e−2β se−4γ(t−s)ds + ct 4γ2 ( e−2γh−1 )2 P ( X0(1)2+ 1 β X0(1) )∫ t 0 e−2β se−4(t−s)γds ≤C2(t,γ,β )h2e−ζ1t where C2 is polynomial in t. By another application of the BDG Inequality, and 82 noting that ‖φ‖= 1, P (|I3|4)= P[(∫ t+h t ∫ e−β sPt+h−sφ(x)dM(s,x) )4] ≤ cP [(∫ t+h t ∫ ( e−β sPt+h−sφ(x) )2 Xs(dx)ds )2] ≤ cP [(∫ t+h t ∫ e−2β sXs(dx)ds )2] ≤ chP [∫ t+h t e−4β sXs(1)2ds ] = che−2β t ∫ t+h t P [ e−2β sXs(1)2 ] ds ≤ che−2β t ∫ t+h t P [ X0(1)2+X0(1)/β ] ds ≤C3(β )h2e−2β t where the second last line follows from the same calculations performed in esti- mating moments of I2. Note that the constant C3 does not depend on t here. Putting the pieces together shows that there exists a function C polynomial in t and a positive constant ζ ∗ such that P [( e−β (t+h)Xt+h(φ)− e−β tXt(φ) )4]≤C(t,γ,β ,d)h2e−ζ ∗t The following is a very useful result of Garsia, Rodemich, and Rumsey [7]. Let Ψ : Rd → R, p : R+→ R be positive, continuous functions. Further, sup- pose Ψ is symmetric about 0 and convex with lim|x|→∞Ψ(x) = ∞ and p(x) is in- creasing with p(0) = 0 . Proposition 3.4.8. If f is a measurable function on [0,1] such that ∫ ∫ [0,1]2 Ψ ( f (t)− f (s) p(|t− s|) ) dsdt = B< ∞ (3.17) 83 then there is a set K of measure 0 such that if s, t ∈ [0,1]\K then | f (t)− f (s)| ≤ 8 ∫ |t−s| 0 Ψ−1 ( B u2 ) d p(u). (3.18) If f is also continuous, then K is in fact the empty set. With this result in hand, we can now bring everything together to prove con- vergence of X̃t . Let d denote the Vasserstein metric on the space of finite measures on Rd . I.e., for µ,ν finite measures, d(µ,ν) = sup φ∈Lip1 ∫ φ(x)d(µ−ν)(x), recalling that Lip1 = { ψ ∈C(Rd) : ∀x,y, |ψ(x)−ψ(y)| ≤ |x− y| ,‖ψ‖ ≤ 1}. Theorem 3.4.9. Suppose X0 (1)< ∞ a.s. and γ > 0. Then on S, d(X̃t ,P∞) a.s.−→ 0, where Pt is the semigroup of an Ornstein-Uhlenbeck process with attraction to 0 at rate γ. Proof. The strategy for this proof is simple: We use Remark 3.4.6 to see that we can lay down an increasingly (exponentially) denser sequence e−β tnXtn(φ) which converges almost surely, and that we can use Lemma 3.4.7 to get a modulus of continuity on the process e−β tnXtn(φ), which then implies that if the sequence is converging, then the entire process must be converging. Assume that P ( K0 (|y0|8)) = P(X0 (|x|8)) < ∞ and P(X0(1)8) < ∞ and ar- gue as in Theorem 3.4.1 in the general case. Let φ ∈ Lip1. Denote e−β tXt by X̊t for the remainder of the proof. Let T > 0, and let Ψ(x) = |x|4 and p(t) = |t|3/4 ( log ( λ t ))1/2 where λ = e4. Let BT (ω) be the constant B that appears in Proposition 3.4.8, with aforementioned functions Ψ and p, for the path X̊Tt(ω), t ∈ [0,1]. 84 Then note that P(BT )≡ P [∫ ∫ [0,1]2 Ψ ( X̊Tt − X̊T s p(|t− s|) ) dsdt ] = ∫ ∫ [0,1]2 P [∣∣X̊Tt − X̊T s∣∣4] |t− s|3 log2 ( λ |t−s| )dsdt ≤ ∫ ∫ [0,1]2 C(T (s∧ t))e−ζ ∗(s∧t)T 2|t− s|2 |t− s|3 log2 ( λ |t−s| ) dsdt = 2T 2 ∫ 1 0 ∫ t 0 C(T s)e−ζ ∗(T s)|t− s|2 |t− s|3 log2 ( λ |t−s| ) dsdt ≤ 2C(T )T 2 ∫ 1 0 ∫ t 0 1 |t− s| log2 ( λ |t−s| )dsdt ≤ C(T )T 2 2e4 where C is the polynomial term that appears in Lemma 3.4.7. Since X̊t is continu- ous, by Garsia-Rodemech-Rumsey, for all s, t ≤ 1, ∣∣X̊Tt − X̊T s∣∣≤ 8∫ |t−s| 0 ( BT u2 ) 1 4 d p(u) ≤ AB 1 4 T |t− s| 1 4 ( log λ |t− s| ) 1 2 where A is a constant independent of T (see Corollary 1.2 of Walsh [22] for this calculation). Rewriting the above, ∣∣X̊t − X̊s∣∣≤ DT |t− s| 14 (log λT|t− s| ) 1 2 ∀s< t ≤ T, (3.19) where DT ≡ A (BT T ) 1 4 . Note that P(D4T ) = A4TC(T ) 2e4 , which is polynomial in T of fixed degree d0 > 1. Let Ω0 be the set of probability 1 such that for all positive integers T Equation (3.19) holds and DT ≤ T d0 for T large enough. To see P(Ω0) = 85 1, use Borel Cantelli: P ( DT ≥ T d0 ) = P ( D4T ≥ T 4d0 ) ≤ P(D 4 T ) T 4d0 ≤ c T 3d0 which is summable over all positive integers T . Suppose ω ∈ Ω0. Let δ T (ω) be such that δ −18 ( log λδ )−1 2 = T d0 . Then for all integral T > T0(ω), and s, t ≤ T with |t− s| ≤ δ , ∣∣X̊t − X̊s∣∣≤ |t− s| 18 . Now let { X̊tn } be a sequence of the form in Remark 3.4.6, with the additional condition that {tn}∩ [k,k+ 1) are evenly spaced within [k,k+ 1) for each k ∈ Z+ (i.e. tn+1− tn = ce−ζk/2 for tn ∈ {tn}∩ [k,k+1)). Evidently X̊tn converges a.s. to a limit X̊∞. Without loss of generality, assume convergence of the sequence on the set Ω0. There exists T1(ω) such that for all T > T1, ce−ζT/2 < δ . Hence, for all t such that T1 ∨ T0 < t ≤ T there exists t ′n ∈ {tn} such that |t− t ′n| < ce−ζbtc/2 < δ and hence ∣∣X̊t − X̊∞∣∣≤ ∣∣X̊t − X̊t ′n∣∣+ ∣∣X̊t ′n− X̊∞∣∣ ≤ ∣∣t− t ′n∣∣ 18 + ∣∣X̊t ′n− X̊∞∣∣ ≤ ce− ζbtc16 + ∣∣X̊t ′n− X̊∞∣∣ . Sending t → ∞ gives almost sure convergence of e−β tXt(φ) to X̊∞ = WP∞(φ) by Theorem 3.4.5, since t ′n→ ∞ with t. Note that this implies for φ ∈ Lip1 X̃t(φ) a.s.−→ P∞(φ) (3.20) since on S, e β t Kt(1) →W−1 a.s. By Exercise 2.2 of [6] on M1(Rd), the space of probability measures on Rd , the Prohorov metric of weak convergence is equivalent to the Vasserstein metric. 86 Note that the class of functions Θ= { ψ : ψ(x) = n ∑ i=1 pi(x)e−qi(x−xi) 2+bi , pi polynomial with rational coefficients,qi ∈Q+,bi ∈Q,xi ∈Qd ,n< ∞ } is a countable algebra of Lipschitz functions that is strongly separating (see p. 113 of [6]). Hence by Theorem 3.4.5(b) of [6], Θ is convergence determining. Since there exists a set S0 ⊂ S with P(S\S0) = 0 such that on S0, Equation (3.20) holds simultaneously for all φ ∈Θ, X̃t(·)→ P∞ (·) in the Vasserstein metric, for ω ∈ S0 because Θ is convergence determining. To drop the dependence on the eighth moment, we argue as in Theorem 3.4.1, where we make use of the Markov Property and the Compact support property for Historical Brownian Motion. Remark 3.4.10. It is possible to show that Theorem 3.4.9 holds for a more general class of superprocesses. That is, if the underlying process has an exponential rate of convergence to a stationary distribution, then the above theorem goes through. One can appeal to, for example, Theorem 4.2 of Tweedie and Roberts [17] for a class of such continuous time processes. Recall that X ′t (·) =Kt (Yt ∈ ·) is the SOU process with attraction to its centre of mass. Theorem 3.4.11. Suppose X ′0 (1)< ∞ a.s. and γ > 0. Then on S, d(X̃ ′t ,P Ȳ∞ ∞ ) a.s.−→ 0 (3.21) where PȲ∞∞ represents the OU-semigroup at infinity, with the origin shifted to Ȳ∞. Proof. This follows almost immediately from Theorem 3.4.9 and the representa- 87 tion given in Remark 3.3.4 (a). Let φ ∈ Lip1, then X̃ ′t (φ) = K̃t (φ (Yt)) = K̃t ( φ ( Zt + γ ∫ t 0 Z̄sds )) = ∫ φ (x+ f (t,ω))dX̃t(dx) where f (t) = γ ∫ t 0 Z̄sds. Remark 3.3.4 (b) gives f (t) = Ȳt − Z̄t , and hence f (t) a.s.−→ Ȳ∞ follows from Theorem 3.4.2. Note that∣∣∣X̃ ′t (φ)−PȲ∞∞ (φ)∣∣∣≤ ∣∣∣∣∫ φ (x+ f (t))dX̃t(dx)−∫ φ (x+ f (∞))dX̃t(dx)∣∣∣∣ + ∣∣∣∣∫ φ (x+ f (∞))dX̃t(dx)−∫ φ (x+ f (∞))dX̃∞(dx)∣∣∣∣ ≤ | f (t)− f (∞)|+d(X̃t , X̃∞) since φ ∈ Lip1. Taking the supremum over φ and the previous theorem give d(X̃ ′t , X̃ ′ ∞)≤ | f (t,ω)− f (∞,ω)|+d(X̃t , X̃∞) a.s.−→ 0. 3.5 The repelling case Much less can be said for the SOU process repelling from its center of mass than in the attractive case. However we can show convergence of the center of mass, provided the rate of repulsion is not too strong. Recall that in the attractive case, this was the first step towards showing the a.s. convergence of the normalized in- teractive SOU process. We finish with some conjectures on the limiting measure for the repelling case. As in the previous section, we will assume that Z0 = Y0 = y0, unless stated otherwise. Theorem 3.5.1. Assuming K0(1)< ∞ a.s., the following hold on S: (a) For 0> γ >−β2 , Ȳt converges almost surely. 88 (b) For 0> γ >−β2 , Z̄t diverges exponentially fast. That is, PX0 ( lim t→∞e γt Z̄t → L 6= 0 ∣∣∣S)= 1. Proof. Assume that P ( K̃ (|y0|2)) < ∞, as in the proof of Theorem 3.4.1. As in that theorem, this condition can be weakened here to just the finite initial mass condition using similar reasoning. For part (a), note that as in Theorem 3.4.2, Ȳt will converge if P([Ȳ ]t)< ∞ and which holds if the following quantity is bounded: P (∫ ∞ 0 |Ys|2− Ȳ 2s eβ s ds;S ) ≤ P (∫ ∞ 0 Y 2s eβ s ds;S ) ≤ c ∫ ∞ 0 1+ s6e−2γs eβ s ds < ∞, by Lemma 3.3.9 and by the conditions on γ . For (b), we require the following Lemma: Lemma 3.5.2. Let −β/2 < γ < 0 and X0 6= 0. For a measure m on Rd , let τa(m) be m translated by a ∈ Rd . That is, τa(m)(φ) = ∫ φ(x+a)m(dx). Then (i) For all but at most countably many a, Pτa(X0) ( eγt Z̄t → L 6= 0|S ) = 1. (3.22) (ii) For all but at most one value of a Pτa(X0) ( eγt Z̄t → L 6= 0|S ) > 0. Proof of Lemma. We first note that, by the correspondence (3.5), we have that Z̄t = Ȳt − γe−γt ∫ t 0 eγsȲsds. 89 Under our hypotheses Ȳt converges by Theorem 3.5.1 (a), and hence lim t→∞e γt Z̄t + ∫ t 0 γeγsȲsds = 0 (3.23) a.s. on S. Therefore, on S, lim t→∞e γt Z̄t exists a.s. (3.24) Note that one can build a solution of (SE)2 with initial conditions given by τa(X0) by seeing that if Yt gives the solution of (SE)2Y0,K , then Yt + a gives the solution of (SE)2Y0+a,K , and that the projection X ′t (·) = ∫ 1(Yt +a ∈ ·)Kt(dy) gives the appropriate interacting SOU process. By (3.23) and (3.24), PX0 ({ eγt Z̄t → L 6= 0 }c ∣∣∣S)= Pτa(X0)(limt→∞∫ t0 eγsȲsds = 0|S ) = PX0 ( lim t→∞ ∫ t 0 eγs Ks (τa(Ys)) Ks(1) ds = 0 ∣∣∣S) = PX0 ( lim t→∞ ∫ t 0 eγs ( a+ Ks(Ys) Ks(1) ) ds = 0 ∣∣∣S) = PX0 ( −a γ + lim t→∞ ∫ t 0 eγsȲsds = 0 ∣∣∣S) . The random variable ∫ ∞ 0 e γsȲsds is finite a.s. and so only a countable number of values a exist with the latter expression positive, implying the first result. The second result also follows as well since the last expression in the above display can be 1 for at most 1 value of a. To complete the proof of Theorem 3.5.1 (b), choose a value a ∈ Rd such that (3.22) holds. By Theorem III.2.2 of [14] and the fact that X0Ps τa (X0)Pt , for all 0 < s ≤ t, for the Ornstein-Uhlenbeck semigroup Pt , we have that for all 0< s≤ t PX0 (Xs+· ∈ ·) Pτa(X0) (Xt+· ∈ ·) . (3.25) 90 By our choice of a, Pτa(X0) ( PX1 ( lim t→∞e γt Z̄t = 0,S )) = 0, holds, and hence by (3.25) we have PX0 ( lim t→∞e γt Z̄t = 0,S ) = PX0 ( PX1 ( lim t→∞e γt Z̄t = 0,S )) = 0. Recalling from (3.24) that limt→∞ eγt Z̄t exists a.s., we are done. Note that for 0 > γ > −β2 , this implies that even if mass is repelled at rate γ , the COM of the interacting SOU process still settles down in the long run. That is, driving Yt away from Ȳt seems to have the effect of stabilizing it. One can think of this as a situation where the mass is growing quickly enough that the law of large numbers overcomes the repelling force. More surprising is that the COM of the ordinary SOU process diverges ex- ponentially fast, even while the COM of the interacting one settles down. This follows from the correspondence Ȳt = Z̄t + γ ∫ t 0 Z̄sds, and the cancellation that occurs in the equation due to the exponential rate of Z̄t . The next lemma shows that Theorem 1 of Engländer and Winter [5] can be reformulated to yield a result for the SOU process with repulsion at rate γ (where γ is taken to be a negative parameter in our setting): Lemma 3.5.3. On S, for the SOU process, X , with repulsion rate −βd < γ < 0 and compactly supported initial measure µ , and any ψ ∈C+c (Rd) ed|γ|t X̃t(ψ) P−→ ξ ∫ Rd ψ(x)dx, where ξ is a positive random variable on the set S. Proof. Note that by Example 2 of Pinsky [15] it is shown that the hypotheses of 91 Theorem 1 of [5] hold for the SOU process with repulsion from the origin at rate 0<−γ < βd . The theorem says that there is a function φc ∈C∞b ( Rd ) such that Xt(ψ) Eµ (Xt(ψ)) P−→ Wξ µ(φc) , (3.26) where W is as in Lemma 3.3.7. Example 2 also shows that for ψ ∈C+c ( Rd ) , lim t→∞e −(β+γd)tEµ (Xt(ψ)) = µ(φc)m(ψ) where m is Lebesgue measure on Rd . Hence, manipulating the expression in (3.26) by using the previous equation and Lemma 3.3.7 gives e|γ|dt X̃t(ψ) P−→ ξW µ(φc) lim t→∞ e|γ|dtEµ (Xt(ψ)) Xt(1) = ξW µ(φc) lim t→∞ e−(β+γd)tEµ (Xt(ψ)) e−β tXt(1) = ξW µ(φc) µ(φc)m(ψ) W . This lemma indicates that on the survival set, when γ < 0, one cannot naively normalize Xt by its mass since the probability measures { X̃t } are not tight. That is, a proportion of mass is escaping to infinity and is not seen by compact sets. Note that the lemma above implies that for Xt , the right normalizing factor is e(β+γd)t . Definition. We say a measure-valued process X̊ undergoes local extinction if for any finite initial measure X̊0 and any bounded A ∈ B(Rd), there is a PX̊0-a.s. finite stopping time τA so that X̊t(A) = 0 for all t ≥ τA a.s. Remark 3.5.4. Example 2 of Pinsky [15] also shows that for γ ≤−β/d the SOU undergoes local extinction (all the mass escapes). Hence for ψ ∈ Cc(Rd), there is no normalization where Xt(ψ) can be expected to converge to something non- trivial. 92 From Remark 3.3.6, one can show that Z̄t = Z̄0+Nt − γ ∫ t 0 Z̄sds, where N is a martingale. Therefore can think of the COM of X , the SOU process repelling from origin, as being given by an exponential drift term plus fluctuations. The correspondence of Remark 3.3.4(b) implies then that Ȳt = Ȳ0+Nt , or in other words, the center of mass of the SOU process repelling from its COM is given by simply the fluctuations. If one fixes a compact set A ⊂ Rd , then for the ordinary SOU process, X , A is exponentially distant from the COM of the process. However with X ′, A will possibly lie in the vicinity of the COM of the process for all time. Therefore one might expect that A is charged by a different amount of mass by X ′t than Xt , and thus we might need to renormalize differently for the two cases. We finish with some conjectures: Conjecture 3.5.5. On the survival set, if X ′0 is fixed and compactly supported, then the following is conjectured to hold: (a) If 0 < −γ < βd then there exists constant β + γd ≤ α < β so that for φ ∈ Cc(Rd), e−αtX ′t (φ) P−→ ν(ψ) where ν is a random measure depending on Ȳ∞. (b) If β/d ≤−γ then X ′t undergoes local extinction. Note that we expect that α < β simply because of the repulsion from the COM built in to the model results in a proportion of mass being lost to infinity. One would expect that the limiting measure ν is a random multiple of Lebesgue measure as in the ordinary SOU process case, due to the correspondence, but it is conceivable that it is some other measure which has, for example, a dearth of mass near the limiting COM. 93 It is difficult to use Lemma 3.5.3 to prove this conjecture as the correspondence becomes much less useful in the repulsive case. The problem is that while the equation ∫ φ(x)dX ′t (x) = ∫ φ ( x+ γ ∫ t 0 Z̄sds ) dXt(x) still holds for t finite, the time integral of Z̄s now diverges. 94 Chapter 4 Conclusions The significance of the above research lies in part with the use of Perkins’ historical calculus to carry out our analyses. This tool has, for the most part, been underuti- lized by the broader mathematical community in the construction and study of interacting measure-valued processes since its introduction by Perkins in the early nineties. We have shown that it can be successfully applied in a transparent man- ner to many problems that lie unsolved in this area. Partially, the importance of research comes from successfully showing that one can determine the equilibrium behavior of a particular interacting measure-valued diffusion (the super Ornstein- Uhlenbeck process attracting to its center of mass). Dawson, Li and Wang in [2] use a scaling limit method to create a superpro- cess in a random medium which has location dependent branching according to any given bounded Borel measurable function. While we did not implement this in Chapter 2, the work there could be extended to include variable branching in the vein of Chapter 4 of Perkins [13], although only for location dependent branching functions that are bounded and twice differentiable in space. It is not clear if this method can be extended to include arbitrary Borel measurable branching functions. However, one could use the methods of [13] to incorporate certain types of inter- active branching, which has not been achieved in any model of this sort thus far. A location dependent branching density is interesting since it is conceivable that particles (like plankton) might give birth to more offspring in some areas of space (ocean) than others. An interactive density is even more interesting since then the 95 rate at which particles branch could vary according to the positions of other par- ticles. This is desirable if, for example, a particle of a particular species is more likely to give birth when there are other particles nearby than when the particle is isolated. Chapter 2 is motivated by plankton dynamics and could find a number of appli- cations in that area. One could go through the oceanographic literature and decide explicitly which coefficients σ1, σ2, and b are most appropriate to model a given population of oceanic phytoplankton. It would be especially interesting to decide what the interactions between the individual plankton should be (and these might vary for different species of plankton). Once a model is decided upon, long term behavior and equilibria could be considered. To do this one may have to overcome some obstacles in the theory of historical calculus, particularly by loosening the restrictions on the types of initial measures admitting solutions to the strong equa- tions (SE)Z0,K . This technical hurdle by itself would be an interesting problem, and once cleared paves the way for considering long-term behavior of different types of measure-valued processes, including those which are immersed in random media. It would perhaps be more realistic to consider colored noise, instead of white noise, in Chapter 2 to model the dependent spatial motion. Colored noise differs from white noise in that there is a spatial correlation structure built in which com- plicates analysis. The first step in this direction might be to prove that given a colored noise independent of a historical process remains a colored noise (orthog- onal to Kt) under the different measures P̂T . In Chapter 3, our most striking results come for the super Ornstein-Uhlenbeck process attracting to its center of mass. Much remains to be shown for the super Ornstein-Uhlenbeck process repelling from its center of mass: for now, a descrip- tion of the limiting behavior of the process remains elusive. One would guess that the limiting measure should be a random multiple of Lebesgue measure since this is the case for the ordinary SOU process, but it is not entirely clear that this is true since the correspondence between the ordinary SOU process and the SOU re- pelling from its COM is no longer useful. The COM of the ordinary SOU process diverges, whereas it converges in the interacting SOU, implying that there might be a qualitative difference in the limiting behavior of the two processes. In regards to the interactive SOU process, there are many directions for future 96 research. First one can look at the aforementioned repelling case, for which little has been established. Secondly, one can consider the critical SOU process attract- ing to its COM (critical in the sense that the branching law associated with the process has mean 1), and condition it to survive indefinitely. The goal then would be to establish similar equilibrium behavior as in the supercritical case for this new process. The conditioned interacting SOU process has properties which are highly de- sirable since physical systems of particles usually behave as if they are critical (and so do not have unchecked population growth) and survive for long periods of time (which is unlike critical processes). Further, this model captures an aspect of bio- logical particle systems where growth is exponential when the population is small and growth is curtailed when the population is large. The coupling found in Chapter 3 between an ordinary SOU and the interacting SOU can be adapted for this new setting and so the key will be to show that the COM of the interacting SOU converges in some sense. Proving this will be difficult for two reasons: We can no longer rely on it being a martingale and so the drift term will need to be controlled. Secondly, the mass generated by a superprocess conditioned to survive is only O(t), making it harder to take advantage of “law of large numbers” type of behavior than in the supercritical case where there is exponential mass growth. A different direction of research is to consider what is known as the Fleming- Viot measure-valued process with the same spatial interaction through the center of mass as considered in Chapter 3. The Fleming-Viot process arises as a scaling limit of a number of discrete models for population genetics (such as the Wright- Fisher model). The mean reversion in this process comes from the fact that in the discrete models, the genetic fitness of each individual is determined in comparison to the fitness of the entire population, which is represented by the mean (COM) of the population. 97 Bibliography [1] D. Dawson. Stochastic Evolution Equations and Related Measure Processes. J. Mult. Anal., 5:1–52, 1975. → pages 2 [2] D. Dawson, Z. Li, and H. Wang. Superprocesses with Dependent Spatial Motion. Elec. J. Prob., 6:1–33, 2001. → pages 6, 13, 35, 40, 95 [3] J. Engländer. The Center of Mass for Spatial Branching Processes and an Application for Self-Interaction. To Appear, 2010. → pages 9, 45 [4] J. Engländer and D. Turaev. A Scaling Limit Theorem for a Class of Su- perdiffusions. Ann. Prob., 30(2):683–722, 2002. → pages 48 [5] J. Engländer and A. Winter. Law of Large Numbers for a Class of Superdif- fusions. Ann. I. H. Poincaré, 42(2):171–185, 2006. → pages 48, 91, 92 [6] S. Ethier and T. Kurtz. 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