Interacting Measure-Valued Diffusions and Their Long-Term Behavior

Interacting Measure-Valued Diffusions and their Long-Term Behavior

Hardeep Gill

B.Sc. Pure Mathematics and Statistics, Calgary, 2005 M.Sc. Mathematics, University of British Columbia, 2007

A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF

Doctor of Philosophy

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 interacting 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 system 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 independence 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 particle 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 attract 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 originally 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 particle 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 position 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 system 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 branching, BPS can be inferred from the properties of the corresponding measure-valued process. A mathematically rigorous result of this type was established independently by Watanabe [25] and later by Dawson [1] for non-interacting BPS, where the associated measure-valued process is a superprocess. The situation for interacting 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 theory 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 particle follows the stochastic process {Yt : t ≥ 0}, which takes values in a space E d (usually E is R ). Also suppose that when the particle dies it branches into k particles 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

{Y α (t),τ(α) ≤ t < d(α) : α ∈ Λ} which will describe the paths of the particles. In particular, for particle α we will 0 insist that Y α (τ(α)) = Y α (d(α0)−), where α0 is the parent of α. This simply says that where the parent α0 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) = ∑ δY α (t)∈A, α∼t 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 process). 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 integral 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

N 1 X (A) = α , (1.1) t ∑ δYt ∈A N α∼t

N where X0 (·) 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, N Markov process X such that Xt −→ Xt , for all t, in distribution with X0 = m. The process X is called a (Y,β,γ)-Dawson-Watanabe superprocess. This proposition 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 N the X 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

N 1 H (B) = α . t ∑ δY·∧t ∈B N α∼t

α Here, for t < τ(α), we let Yt 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

N Ht −→ 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, Z Xt (A) = 1(yt ∈ A)Ht (dy).

One of the special features of superprocesses that results from the independence 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 −R φdX −R V φdX E e t = e t 0 (1.2)

4 where Vs ≡ Vsφ is the solution to

∂V γ s = AV + βV − V 2, (1.3) ∂s s s 2 s 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 populations 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 connection 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’ distribution 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, α as in Proposition 1.1.1. Under this method the particle α follows a path Yt on its life time, where Y α solves the following stochastic differential equation:

Z t Z t Z t Z α α α α α α Yt = Yτ + σ1(Ys )dBs + b(Ys )ds + σ2(Ys ,ξ)dW(s,ξ) (1.4) τ τ τ where σi and b are coefficients satisfying some reasonable conditions and τ ≡ α τ(α). Here the {B }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 N random (spatial) direction. Then for a set A in Euclidean space, Wt (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 Vt φ 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 measurable. Then deﬁne (SE)

Z t Z t Z t Z (a) Zt = Z0 + σ1(Xs,Zs)dy(s) + b(Xs,Zs)ds + σ2(Xs,Zs,ξ)dW(s,ξ) 0 0 0 Rm Z (b) Xt (A) = 1(Zt (y) ∈ A)Kt (dy)

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 as the natural analogue to (1.4). One major difference is that (SE)(a) includes interactions 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) solution 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 support 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 systems 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 = −γVds + 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

Z t Z t α α ¯ α α Yt = Yτ(α) + γ (Y(s) −Ys )ds + dBi (s), (1.5) τ(α) τ(α) where ¯ 1 α Yt = ∑ Yt #{α ∼ t} α∼t is the center of mass and {Bα : α ∈ Λ} is an independent family of Brownian motions. This constructs a branching modiﬁed-OU system Z as follows:

Z (A) = α . (1.6) t ∑ δYt ∈A α∼t

The particles in the system Z exhibit attraction to its COM when γ > 0 and repulsion when γ < 0. The motivation for the model came from a paper of Englander¨ [3] where he constructs a branching particle system Z0 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. Englander¨ then proves various theorems about the long term behavior of the particle system and shows that it achieves a natural equilibrium. He proceeds by ﬁrst showing that the center of mass Y¯t converges as t approaches inﬁnity and then that d for φ : R → R nice, Z Z −n 0 Y¯∞ 2 φ(x)Zn(dx) −→ φ(x)P (dx), (1.7)

¯ as n goes to inﬁnity, almost surely, where PY∞ is a Gaussian random variable cen-

9 tered at the limiting value of the COM, Y¯∞. 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 supercritical 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 supercritical 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 following 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(Y¯s −Ys)ds + dys Z (b) Xt (A) = 1(Yt (y) ∈ A)Kt (dy)

d where A ⊂ R and the COM is deﬁned as R xX (dx) ¯ s Ys = R . 1Xs(dx)

We proceed to show that Y¯t converges almost surely and use this fact to prove the main theorem:

x Theorem 1.3.1. If P∞ is the stationary distribution of an Ornstein-Uhlenbeck process with attraction to x at rate γ > 0, then on the survival set

Xt (·) a.s. ¯ −→ PY∞ (·), Xt (1) as t → ∞.

The convergence obtained here is in fact even stronger than the one Englander¨

10 showed for (1.7) since here we have convergence in measure. The methods used to get this result differ considerably from those of Englander¨ due to the generalized setting being considered. The novelty of this work is that it is one of few cases where we can explicitly 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:

Z t (SE)(a) Zt = Z0 + σ1(Xs,Zs)dy(s) 0 Z t Z Z t + σ2(Xs,Zs,ξ)dW(s,ξ) + b(Xs,Zs)ds 0 Rm 0 Z d (b) Xt (A) = 1(Zt (y) ∈ A)Kt (dy) ∀A ∈ B(R ).

d d d×d d d The coefficients are such that σ1 : MF (R ) × R → R , σ2 : MF (R ) × R × m d×d d d d d R → R and b : MF (R ) × R → R , where MF (R ) denotes the space of all d non-negative finite measures on R . K is a historical super-Brownian motion on d m some space, y is a path in R , and W is a d-dimensional white noise (on R+ × R ) 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 d d space of paths in R , while X takes values in the space of paths in MF (R ).

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). d t t Let C = C(R+,R ) and C be its Borel σ-ﬁeld. Also deﬁne C = {y : y ∈ C}

13 s t and Ct = σ(y ,s ≤ t,y ∈ C) where y = y·∧t . Deﬁne MF (C) to be the space of ﬁnite 0 0 0 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 E0 and the space of bounded continuous functions from E to E0, respectively.

Notation. For a measure µ on a space E and a measurable function f : E → R, let µ( f ) = R 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 Bˆt = B(· ∧ t) be the path-valued process associated with B, taking values in Ct . Then for φ ∈ bC (bounded and C s,y measurable), if s ≤ t let Ps,t φ(y) = E (φ(Bˆt )), where the right hand side denotes expectation at time t given that until time s, Bˆ follows the path y. We can now introduce the weak generator, Aˆ, of Bˆ. If φ : R+ ×C → R we say φ ∈ D(Aˆ) if and only if φ is bounded, continuous and (Ct )-predictable, and for some Aˆsφ(y) with the same properties as φ,

Z t φ(t,Bˆ) − φ(s,Bˆ) − Aˆrφ(Bˆ)dr,t ≥ s, s

s is a (Ct )-martingale under Ps,y for all s ≥ 0,y ∈ C . d Then if m ∈ MF (R ), we say K satisﬁes the historical martingale problem, (HMP)m, if and only if K0 = m a.s. and

Z t ∀φ ∈ D(Aˆ), Mt (φ) ≡ Kt (φt ) − K0(φ0) − Ks(Aˆsφ)ds 0 Z t 2 is a continuous (Ft )-martingale with hM(φ)it = Ks(φs )ds ∀t ≥ 0, a.s. 0

The deﬁnition of Mt (·) can be extended to form an orthogonal martingale measure 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

Z t 2 Ks(ψs )ds < ∞ ∀ t ≥ 0, (2.1) 0 then there exists a continuous local martingale Mt (ψ) with quadratic variation

14 R t 2 given by hM(ψ)it = 0 Ks(ψs )ds. If the expectation of the term in (2.1) is ﬁnite, 2 then Mt (ψ) is an L martingale. In this paper, we will require m(·) ≡ P(K0(·)) to be a ﬁnite, positive measure d on R . Recall that for a measure space (E,E), the universal completion of E, denoted E∗, is given by ∗ E = ∩µ E¯ µ , where E¯ µ denotes the completion of E with respect to the measure µ and the inter- section is over all probability measures µ on E.

ˆ ˆ ˆ ˆ ∗ Deﬁnition. Let (Ω,F,Ft ) = (Ω ×C,F × C,Ft × Ct ). Let Ft denote the universal completion of Fˆ t . 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 ˆ ˆ ˆ measure PT on (Ω,F) given by

ˆ −1 PT (A × B) = P(1AKT (B))m(1) for A ∈ F,B ∈ C.

ˆ ˆ We denote sample points in Ω by (ω,y). Hence, under PT , ω has law KT (1)dP and conditionally on ω, y has law KT (·)/KT (1).

ˆ Note that here PT is a probability measure, since

ˆ −1 −1 PT (1) = P(KT (1))m(1) = P(K0(1))m(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 ∈ Fˆ T , bounded. Then

Z t Z Kt (ψ) = KT (ψ) + ψ(y)dM(s,y) ∀t ≥ T P-a.s. T

15 ˆ ˆ Proposition 2.2.2. If T ∈ Tb then under PT , y is an (Ft )-adapted Brownian motion stopped at T.

m Suppose that {W(t,ξ),ξ ∈ R ,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 ∈ m n×d R t R C ( , ), let W ( ) = m ( )dW(s, ). This deﬁnes the associated white c R R t φ 0 R φ ξ ξ noise process of W, and hence we will identify the two in what follows. The next ˆ ˆ theorem will be useful in proving that W is also an (Ft )-adapted PT Brownian sheet for each T ∈ Tb.

Theorem 2.2.3. Suppose N is a real-valued, square-integrable, (Ft )-adapted martingale that is independent of K. Then for each T ∈ Tb, N is an (Fˆ t )-adapted ˆ j ˆ martingale under PT and hN,y it = 0 ∀t ≥ 0 PT -a.s ∀ j ≤ d.

Proof. Let A ∈ Cs and B ∈ Fs. Then

ˆ PT [(Nt − Ns)1A1B]

= P[(Nt − Ns)KT (A)1B]

= P[1B1T≤sKT (A)E(Nt − Ns|Fs)] + P[(Nt − Ns)1B1T>sKT (A)] Z T Z = P (Nt − Ns)1B1T>s Ks(A) + 1AdM(r,y) s Z T∧t Z = P (Nt − Ns)1B1T>s Ks(A) + 1AdM(r,y) s 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

Z T∧t Z ˆ PT [(Nt − Ns)1A1B] = P (Nt − Ns)1B0 1AdM(r,y) s Z T∧t Z = P (NT∧t − Ns)1B0 1AdM(r,y) s

0 where B = B ∩ {T > s} ∈ Fs. Now, if we show that Nt Mt (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,

Z t Z t Nt Mt (ψ) = NsdMs(ψ) + Ms(ψ)dNs + hN,M(ψ)it 0 0 Z t Z t = NsdMs(ψ) + Ms(ψ)dNs 0 0 since hN,M(ψ)it = 0 by independence of N and M. It also follows from the independence 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- ˆ tingale as well. Hence Nt Mt (ψ) is a martingale and PT [(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,

ˆ PT (N ∈ A,y ∈ B) = P(1N∈AKT (1y∈B))

= P(1N∈A)P(KT (1y∈B)) ˆ ˆ = PT (N ∈ A)PT (y ∈ B).

ˆ j ˆ Hence, N and y are independent under PT . This implies that hN,y i = 0,PT -a.s. ˆ In the general case where T ∈ Tb, independence of N and y under PT no longer j ˆ necessarily holds. Let Z ≡ hN,y i. Then under PT , for u > T ﬁxed, ˆ PT sup|Zs| ∧ 1 = P KT sup|Zs| ∧ 1 s≤T s≤T Z u Z = P Ku sup|Zs| ∧ 1 − sup|Zs| ∧ 1dM(r,y) s≤T T s≤T = 0

ˆ In the second line, we have noted that sups≤T |Zs| ∧ 1 is bounded, FT -measurable ˆ and applied Proposition 2.2.1. To get the third line, we use the fact that Z = 0, Pu- a.s. by the special case considered above and that the second term is a martingale. j ˆ Hence hN,y it = 0 ∀t ≥ 0 PT -a.s. ∀T ∈ Tb (note that we have used the property ˆ T that under PT , y = y a.s. implicitly here).

17 m m Let R denote the Borel σ-algebra of R . The following proposition is needed to complete the proof.

m m Proposition 2.2.4. Let M be an orthogonal martingale measure on (R ,R ) 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].

m ˆ Corollary 2.2.5. For each T ∈ Tb, {W(t,ξ),ξ ∈ R ,t ≥ 0} is an (Ft )-adapted ˆ ˆ Brownian sheet under PT . Furthermore, W and y are orthogonal under PT ; that j m d ˆ is, hW(φ),y it = 0 ∀φ ∈ Cc(R ,R ),t ≥ 0 PT -a.s. ∀T ∈ Tb for all j ≤ d.

1 d ˆ Proof. Let Wt (A) = (Wt (A),...,Wt (A)). We simply need to verify that under PT , i i {Wt } are independent white noises. Each Wt is an orthogonal martingale measure ˆ m i i i under PT since for disjoint bounded sets A,B ∈ R , Wt (A) and Wt (A)Wt (B) are ˆ square-integrable (Ft )-martingales with respect to P, hence also (Ft )-martingales ˆ i with respect to PT by Theorem 2.2.3. The map t 7→ Wt (A) is continuous since i ˆ i Wt (A)(ω,y) deﬁned on Ω is equal to Wt (A)(ω) on Ω, for which the analogous ˆ map is continuous and PT |Ω is absolutely continuous with respect to P. i 2 ˆ By noting that Theorem 2.2.3 implies Wt (A) − tν(A) is an (Ft )-martingale, where ν is the m-dimensional Lebesgue measure, we see that the covariance mea- i sure is deterministic. Hence each Wt is a white noise process by Proposition 2.2.4. i j Then by a ﬁnal use of Theorem 2.2.3, we see that Wt (A)Wt (A) − tν(A)δi, j is an i j (Fˆ t )-martingale where δi, j is Kronecker’s δ. Therefore W (A) and W (A) are independent 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 n×d 2 2 n For a matrix A ∈ R , let ||A|| = ∑i, j Ai j. Note for a vector x ∈ R , we will use |x|2 instead to denote the same quantity.

ˆ n×d ˆ ∗ Deﬁnition. We say f ∈ D1(n,d) iff f : R+ × Ω → R is (Ft )-predictable and

Z t 2 k f (s,ω,y)k ds < ∞ Kt − a.a. y ∀t ≥ 0,P a.s. 0

m ˆ n×d ˆ ∗ We say g ∈ D2(m,n,d) iff g : R+ × R × Ω → R is (Ft )-predictable and

Z t Z 2 kg(s,ξ,ω,y)k dξds < ∞ Kt − a.a. y ∀t ≥ 0,P a.s. 0 Rm ˆ Definition. If X,Y : R+ × Ω → E, we say X = YK-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. ˆ ˆ Since with respect to each measure PT ,T ∈ Tb, {W(s,ξ),0 ≤ s} is an (Fs)- adapted Brownian sheet, we may define the stochastic integral of a function f with R t R 2 d k f ( ,s, ,y)k d ds < , ˆ -a.s. as in Walsh [22]. However as this integral 0 R ξ ω ξ ∞ PT ˆ R t R depends on T, we denote it by PT - 0 f (s,ξ,ω,y)dW(s,ξ). ˆ R t Similarly, define PT - 0 g(s,ω,y)dy(s) to be the stochastic integral g with re- ˆ spect to y under PT . 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 (Fˆ t )-predictable, K-a.e. continuous process I(g,t,ω,y) such that

Z t ˆ ˆ I(g,t ∧ T,ω,y) = PT - g(s,ω,y)dy(s) for all t ≥ 0,PT -a.s. for all T ∈ Tb. 0

0 ˆ ∗ Moreover, I is unique. That is, if I is an (Ft )-predictable process satisfying the above, then I(g,s,ω,y) = I0(g,s,ω,y) K-a.e.

ˆ R t R We can prove a similar theorem for PT - 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 u 2 m Example 2.2.7. Consider f (s,ξ,ω,y) = y (s)η(ξ)1(u,v](s) with η ∈Cc (R ). Note ˆ T that under PT , y = y a.s., and hence

Z t Z ˆ T PT - f (s,ξ,ω,y)dW(s,ξ) = y (u)(Wv∧t (η) −Wu∧t (η)). 0

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 ˆ when integrated with respect to W under the different measures PT when t > T. The next theorem is the analogue of Proposition 2.2.6 and shows how the stochastic ˆ integrals of W are related under different measures PT .

n ˆ Theorem 2.2.8. (a) If f ∈ D2(m,n,d), there is an R -valued (Ft )-predictable process J( f ,t,ω,y) such that for all T ∈ Tb

Z t∧T Z ˆ ˆ J( f ,t ∧ T,ω,y) = PT - f (s,ξ,ω,y)dW(s,ξ) ∀t ≥ 0,PT -a.s. 0 Rm

0 ˆ ∗ 0 (b) If J ( f ) is an (Ft )-predictable process satisfying (a), then J( f ,s) = J ( f ,s) K-a.e.

(d) (Dominated Convergence) For any N > 0, if fk, f ∈ D2(m,n,d) satisfy

Z N Z 2 lim P KN k fk(s,ξ) − f (s,ξ)k dξds > ε = 0,∀ε > 0, k→∞ 0 Rm

then, 2 lim P supKt supkJ( fk,s) − J( f ,s)k > ε = 0 ∀ε > 0. k→∞ t≤N s≤t

20 (e) For any S ∈ Tb if fk, f ∈ D2(m,n,d) satisfy

Z S Z 2 lim P KS k fk(s,ξ) − f (s,ξ)k dξds = 0, k→∞ 0 Rm

then, 2 P supKt supkJ( fk,s) − J( f ,s)k → 0 as k → ∞. t≤S s≤t The proof of Proposition 2.2.6 can be easily adapted to prove this theorem. We R t R will at times write 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 ˆ ∗ (Ft )-predictable process S( f ,g) satisfying

Z t∧T Z t∧T Z ˆ ˆ S( f ,g,t ∧ T,ω,y) = PT - f (s,y)dy(s) + PT - g(s,ξ)dW(s,ξ) (2.2) 0 0

ˆ ∀t ≥ 0,PT -a.s., then S( f ,g) = I( f ) + J(g),K-a.e.

Proof. Deﬁne Z L(t,ω) = supkI( f ,s) + J( f ,s) − S( f ,g,s)k ∧ 1Kt (dy). (2.3) s≤t

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 ˆ P(L(T,ω)) = PT supkI( f ,s) + J( f ,s) − S( f ,g,s)k ∧ 1 s≤T = 0

ˆ R s∧T ˆ R s∧T R since S( f ,g,T ∧ s) = PT - 0 f (s,y)dy(s) + PT - 0 g(s,ξ)dW(s,ξ), ˆ R s∧T I( f ,s ∧ T) = PT - 0 f (s,ω,y)dy(s) by Proposition 2.2.6 and since J(g,s ∧ T) = ˆ R s∧T R PT - 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 n ˆ ˆ Notation. If X(t) = (X1(t),...,Xn(t)) is an R -valued process on (Ω,F) and µ ∈ R MF (C), let µ(Xt ) = (µ(X1(t)),..., µ(Xn(t))) where µ(Xi(t)) = Xi(t,ω,y)µ(dy). Also let

Z t Z Z t Z Z t Z X(s)dM(s,y) = X1(s)dM(s,y),..., Xn(s)dM(s,y) 0 0 0 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 Ito’sˆ Lemma.

R 2 Theorem 2.2.10. If f ∈ D2(m,n,d) and sups,ω,y k f (s,ξ,ω,y)k dξ < ∞ then

Kt (J( f ,t)) Z t Z Z t Z = Ws( f (s))dM(s,y) + Ks( f (s))dW(s,ξ) ∀t ≥ 0 P-a.s. (2.4) 0 0

2 and each integral on the right is a continuous L (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)

∞ m where φ1 ∈ bFu,φ2 ∈ bCu,φ3 ∈ Cc (R ) and 0 ≤ u < v. Then,

Z t Z Kt f (s,ξ,ω,y)dW(s,ξ) = Kt (φ1(ω)φ2(y)(Wv∧t (φ3) −Wu∧t (φ3))) 0

= φ1(ω)(Wv∧t (φ3) −Wu∧t (φ3))Kt (φ2) Z t Z = φ1(ω)φ3(ξ)1(u,v](s)dW(s,ξ)Kt (φ2) 0 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 hW(φ3),K(φ2)it = 0 (by independence), to get

Z t Z Z t Z Kt f (s,ξ)dW(s,ξ) = φ1(ω)φ3(ξ)1(u,v](s)Ks(φ2)dW(s,ξ) 0 0 Z t Z s Z + φ1(ω)φ3(ξ)1(u,v](r)dW(r,ξ)dKs(φ2) u u Z t Z = Ks( f (s,ξ))dW(s,ξ) 0 Z t Z s Z Z + φ1(ω)φ3(ξ)1(u,v](r)dW(r,ξ) φ2(y)dM(s,y) 0 0 Z t Z Z t Z Z s Z = Ks( f (s))dW(s,ξ) + f (r,ξ,ω,y)dW(r,ξ) dM(s,y). 0 0 0

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 R φ2(y)dM(s,y) is just dMs(φ2). Note that for these simple functions f , we need only look at the square function R t R 2 of 0 Ks( f (s,ξ))dW(s,ξ) to show that it is an L martingale. That is, since φ1φ2 ∞ m is bounded and φ3 ∈ Cc (R ),

Z t Z Z t Z 2 2 2 P Ks( f (s,ξ)) dξds ≤ cP φ3(ξ) Ks(1) dξds . 0 0

This expectation is ﬁnite because Kt (1) is Feller’s branching diffusion. R t R 2 To show that 0 J( f ,s)dM(s,y) is an L martingale consider

Z t Z t Z s Z 2! 2 2 P Ks J( f ,s) ds = P Ks φ2 φ1φ31(u,v]dW(r,ξ) ds 0 0 0 Z t Z s Z 2 ≤ c P Ks (1)P (φ1φ31(u,v]) dξdr ds 0 0 < ∞.

2 Hence the sum of these two quantities is also an L martingale. Since Mt (φ2) is R t R R t R continuous, 0 J( f ,s)dM(s,y) is continuous. As 0 Ks( f (s,ξ))dW(s,ξ) is con- R t R tinuous so is Kt 0 f (s,ξ,ω,y)dW(s,ξ) . Suppose the statement of the theorem holds for a sequence of (Fˆ t )-predictable

23 ˆ ∗ processes fk and f is an (Ft )-predictable process such that fk → f pointwise, R 2 R 2 supk k fk(ξ)k dξ ∨ k f (ξ)k dξ < ∞ and

ZZ N Z 2 lim P ( fk(s,ξ) − f (s,ξ)) dξdsKN(dy) = 0 ∀N ∈ N. (2.6) k→∞ 0

ˆ Since J( f ,s) is a White noise integral with respect to Ps, consider for each N ∈ N,

Z N Z 2 P(hM (J( f ,s) − J( fk,s))iN) = P (J( fk,s) − J( f ,s)) Ks(dy)ds 0 Z N ˆ 2 = Ps (J( fk,s) − J( f ,s)) ds 0 Z N Z s Z ˆ 2 = Ps ( fk(r,ξ) − f (r,ξ)) dξdr ds 0 0 Z N Z s Z ˆ 2 = PN ( fk(r,ξ) − f (r,ξ)) dξdr ds 0 0 where in the second line we have used Fubini’s theorem and the deﬁnition of the ˆ Campbell measure Ps. In the third line we make use of the Itoˆ isometry and then ﬁnally we use Remark V.2.5 (d) of [14] in the last line. The last expression now goes to zero as k → ∞ by (2.6) (and Fubini’s theorem). This shows that R t R 2 0 J( f ,s,y)dM(s,y) is a continuous L martingale. Now by the Dominated Con- vergence Theorem, ∀t as k → ∞,

Z t Z 2 P hJ(Ks( f (s))) − J(Ks( fk(s)))it = P Ks( f (s,ξ) − fk(s,ξ)) dξds 0 → 0,

R t R 2 and so 0 Ks( f (s,ξ))dW(s,ξ) is a continuous L martingale as well.

By using Theorem 2.2.8(e) with S = N and (2.6), we get

P supKt (|J( fk,t) − J( f ,t)|)→ 0 as k → ∞,∀N ∈ N. t≤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].

ˆ n Theorem 2.2.11 (Ito’sˆ Lemma). Let Z0 be F0-measurable and take values in R . n ˆ ∗ Let f ∈ D1(n,d), g ∈ D2(m,n,d), h an R -valued (Ft )-predictable process and 1,2 n ψ ∈ Cb (R+ × R ). Assume

Z t Z t Z 2 2 Ks(k fsk + |hs|)ds + Ks kg(s,ξ)k dξ ds < ∞, ∀t a.s., (2.7) 0 0 and let

Zt (ω,y) = Z0(ω,y) (2.8) Z t Z t Z Z t + f (s,ω,y)dy(s) + g(s,ξ,ω,y)dW(s,ξ) + h(s,ω,y)ds, 0 0 0 then

Z Z ψ(t,Zt )Kt (dy) = ψ(0,Z0)dK0(y) Z t Z Z t Z + ψ(s,Zs)dM(s,y) + Ks(∇ψ(s,Zs)g(s,ξ))dW(s,ξ) 0 0 ! Z t ∂ψ 1 n n + Ks (s,Zs) + ∇ψ(s,Zs) · hs + ∑ ∑ ψi, j(s,Zs)ai j(s) ds, (2.9) 0 ∂s 2 i=1 j=1 where ∇ψ and ψi j are the gradient and second order partial derivatives in the spatial variables and a ≡ f f ∗ + R 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 R ψ(t,Zt )Kt (dy) is a continuous (Ft )-semimartingale.

Proof. For now assume that k f k and |h| are bounded and that R 2 ¯ sups,y kg(s,ξ,ω,y)k dξ < ∞. Let T ∈ Tb and Zt = (t,Zt ). Using the classical ˆ Ito’sˆ Lemma we have, PT -a.s., ∀t ≥ 0

25 ψ(Z¯T∧t ) − ψ(Z¯0) = Z T∧t Z T∧t Z T∧t ¯ ∂ψ ¯ 1 ¯ i j ∇ψ(Zs) · dZs + (Zs)ds + ∑ ψi j(Zs)dhZ ,Z is 0 0 ∂s 2 i, j≤n 0 Z T∧t Z T∧t Z ˆ ˆ = PT - ∇ψ(Z¯s) f (s) · dy(s) + PT - ∇ψ(Z¯s)g(s,ξ) · dW(s,ξ) 0 0 Z T∧t n n ! ¯ ∂ψ ¯ 1 ¯ + ∇ψ(Zs) · h(s) + (Zs) + ∑ ∑ ψi, j(Zs)ai j(s) ds. 0 ∂s 2 i=1 j=1

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 b˜(s) denote the term inside the last integral above, we see that

Z t S(t) = ψ(Z¯t ) − ψ(Z¯0) − b˜(s)ds 0

ˆ ∗ is an (Ft )-predictable process satisfying (2.2). Hence we apply Corollary 2.2.9 (with f and g replaced by ∇ψ(Z¯) f and ∇ψ(Z¯)g respectively) to get

Z t ψ(Z¯t ) = ψ(Z¯0) + I(∇ψ(Z¯) f ,t) + J(∇ψ(Z¯)g,t) + b˜(s)ds. (2.10) 0

Since |b˜|,k∇ψ(Z¯) f k and R k∇ψ(Z¯)g(ξ)k2dξ are all bounded and ˆ ∗ (Ft )-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,

Z Z Z t Z ψ(Z¯t )Kt (dy) = ψ(Z¯0)K0(dy) + ψ(Z¯0)dM(s,y) 0 Z t Z Z t Z + I(∇ψ(Z¯) f ,s)dM(s,y) + J(∇ψ(Z¯)g,s)dM(s,y) 0 0 Z t Z Z s Z t Z + b˜(r)dr dM(s,y) + Ks(∇ψ(Z¯s)g(s,ξ))dW(s,ξ) 0 0 0 Z t + Ks(b˜(s))ds 0 Z Z t Z = ψ(Z¯0)K0(dy) + ψ(Z¯s)dM(s,y) 0 Z t Z Z t + Ks(∇ψ(Z¯s)g(s,ξ))dW(s,ξ) + Ks(b˜(s))ds. 0 0 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 ˆ ∗ R k 2 k k and (Ft )-predictable and such that kg (ξ)k dξ is bounded and f → f ,g → g and hk → h pointwise (by, for example, truncating f ,g and h) with k f kk ≤ k f k,kgkk ≤ kgk and |hk| ≤ |h|. Therefore, by the Dominated Convergence Theo- rem,

Z t Z k 2 k k 2 Ks k fs − fsk + |hs − hs| + kgs − gsk dξ ds → 0 ∀t ≥ 0 a.s. (2.11) 0

Using (2.7) choose Sn ∈ Tb, Sn ↑ ∞, a.s. such that

Z Sn Z Sn Z 2 2 Ks k fsk + |hs| ds + Ks kgsk dξ ds ≤ n a.s. 0 0

Let Zk be as in (2.8) except with f ,g,h replaced by f k,gk,hk respectively. Using K (R t |h |ds) < n Lemma V.3.3(b) of [14] we get that supt≤Sn t 0 s ∞ for all , 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 Z t k P sup Kt |hs − hs|ds →0 as k → ∞ ∀n. (2.12) t≤Sn 0

27 Using Proposition V.3.2(e) and Lemma V.3.3(b) of [14] gives k 2 P sup Kt supkI( f ,s) − I( f ,s)k → 0 as k → ∞ ∀n. (2.13) t≤Sn s≤t

Similarly, using Theorem 2.2.8 above and Lemma V.3.3(b) of [14] gives k 2 P sup Kt supkJ(g ,s) − J(g,s)k → 0 as k → ∞ ∀n. (2.14) t≤Sn s≤t

Together, (2.12), (2.13) and (2.14) give k P supKt sup|Z (s) − Z(s)| →0 as k → ∞ ∀T > 0. (2.15) t≤T s≤t

Since we have that (2.9) holds for (Zk, f k,gk,hk) in place of (Z, f ,g,h), the boundedness 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

Z Xt (A) = 1(Zt (ω,y) ∈ A)Kt (dy)

n deﬁnes an a.s. continuous (Ft )-predictable MF (R )-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 : d d d×d d d m d×d d d MF (R ) × R → R , σ2 : MF (R ) × R × R → R , b : MF (R ) × R → d d d R t R and Z0 : R → R be Borel maps. If 0 σ1(Xs,Zs)dy(s) is the integral I and R t R 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:

Z t (a) Zt (ω,y) = Z0(y0) + σ1(Xs,Zs)dy(s) 0 Z t Z Z t (SE)Z0,K + σ2(Xs,Zs,ξ)dW(s,ξ) + b(Xs,Zs)ds K-a.e. 0 0 Z d (b) Xt (ω)(A) = 1(Zt (ω,y) ∈ A)Kt (dy) ∀A ∈ B(R ),∀t ≥ 0 a.s.

ˆ ∗ d (X,Z) is a solution of (SE)Z0,K iff Z is an (Ft )-predictable R -valued process and d X is an (Ft )-predictable MF (R )-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 : d d d R → R : kφk∞ ≤ 1,|φ(x) − φ(z)| ≤ |x − z| ∀x,z ∈ R } and for µ,ν ∈ MF (R ), d denote the Vasserstein metric on MF (R ) by

d(µ,ν) = sup |µ(φ) − ν(φ)|. φ∈Lip1

Let σ1,σ2 and b be such that there is an increasing function L : R+ → R+ with

0 0 0 0 Lip (a) kσ1(µ,z) − σ1(µ ,z )k + |b(µ,z) − b(µ ,z )| 0 0 0 0 d 0 d ≤ L(µ(1) ∨ µ (1)) d(µ, µ ) + |z − z | ∀µ, µ ∈ MF (R ),z,z ∈ R Z 0 0 2 (b) kσ2(µ,z,ξ) − σ2(µ ,z ,ξ)k dξ

≤ L(µ(1) ∨ µ0(1))[d(µ, µ0)2 + |z − z0|2] Z 2 (c) sup(kσ1(0,z)k + |b(0,z)|) < ∞ and sup kσ2(0,z,ξ)k dξ < ∞. z z

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

Z 2 sup kσ1(µ,z)k + kσ2(µ,z,ξ)k dξ + |b(µ,z)| ≤ α(µ(1)) (2.16) z

Deﬁne TN = inf{t : Kt (1) ≥ N} ∧ N and

29 d S1 = {X : R+ × Ω → MF (R ) : X is (Ft )-predictable, a.s. continuous and

Xt (1) ≤ N ∀t < TN ∀N ∈ N} ˆ d ˆ ∗ S2 = {Z : R+ × Ω → R : Z is (Ft )-predictable and continuous K-a.e.}

S = S1 × S2.

Let

Φ2(X,Z)(t,y) ≡ Z˜t (y) Z t Z t Z Z t = Z0(y0) + σ1(Xs,Zs)dys + σ2(Xs,Zs,ξ)dW(s,ξ) + b(Xs,Zs)ds 0 0 0 Z Φ1(X,Z)(t)(·) ≡ X˜t (·) = 1(Z˜t (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 (X˜ i,Z˜i) = Φ(Xi,Zi) for i = 1,2. If there exist universal constants cN such that for any stopping time T ≤ TN,

Z ˆ ˜1 ˜2 2 ˜1 ˜2 2 PT sup |Z (s) − Z (s)| ∧ 1 ≤ cN |Z0 − Z0 | ∧ 1dm (2.17) s≤t∧T Z t∧T Z t∧T 1 2 2 ˆ ˜1 ˜2 2 + cN P d(Xs ,Xs ) ∧ 1ds + PT |Z (s) − Z (s)| ∧ 1ds , 0 0

∀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 i i as above and T ∈ Tb with T ≤ TN. Let (X˜ ,Z˜ ) be as above, for i = 1,2. Using ﬁrst Doob’s strong L2 inequality and then successive applications of the Cauchy-

Z t∧T # 2 1 2 1 2 2 2 + NL(N) (d(Xs ,Xs ) + |Zs − Zs |) ∧ α(N) ds , (2.18) 0 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 i that Xs(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

ˆ 1 2 2 PT sup |Z˜ (s) − Z˜ (s)| ∧ 1 s≤t∧T Z Z t∧T 1 2 2 2 2 2 1 2 2 ≤ c |Z0 − Z0 | dm + cN L(N) α(N) P d(Xs ,Xs ) ∧ 1ds 0 Z t∧T 2 2 ˆ 1 2 2 + cNL(N) α(N) PT |Zs − Zs |) ∧ 1ds , 0 where we have used Proposition 2.2.1 to handle the ﬁrst term and then factored 2 2 2 2 out the α(N) term inside the integral in (2.18). Letting cN = N L(N) α(N) 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,

0 0 0 0 kσ2(µ,z,ξ) − σ2(µ ,z ,ξ)k ≤ c0(d(µ, µ ) + |z − z |) and Z 2 sup kσ2(0,z,ξ)k dξ < ∞. z 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| m represents the m-dimensional volume of A ⊂ R ).

(ii) σ2(µ,z,ξ) has no measure dependence and has ﬁrst and second derivatives α in z. Also, there exists M > 0 such that supz k∂ (σ2)(z,·)k2 < 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, sup k ∂ σ (z,·)k z ∂zi 2 1 < M.

32 R n nd d+nd+m d×d (iv) σ2(µ,z,ξ) = f (z,x,ξ)dµ (x) where x ∈ R and f : R → R such that f is bounded, twice differentiable in x and z and there exists M > 0 with α supz,x k∂ f (z,x,·)k2 < M for all multi-indices α with |α| ≤ 2. Additionally 2 m assume that there exists an L function c : R → R+ such that for all z, the function f (z,·,ξ) is Lipschitz with constant c(ξ).

n+d+m d×d (v) σ2(µ,z,ξ) = f (µ(φ1),..., µ(φn),z,ξ) where f : R → R 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 ξ 2 ξ g (z) = kσ2(z,ξ) − σ2(z0,ξ)k . Since σ2 is twice differentiable, so is g . Note ξ ξ that for each y, g has a minimum when z = z0 and that g (z0) = 0. Hence, by looking at the Taylor expansion we get

Z 1 ξ 2 α α ξ g (z) = ∑ (z − z0) (1 −t)∂ (g )(z0 +t(z − z0))dt |α|=2 α! 0 and, after letting z(t) = z0 +t(z − z0) we have

Z ZZ 1 ξ 2 α ξ g (z)dξ ≤ c|z − z0| ∑ (1 −t)k∂ (g )(z(t))kdtdξ α=2 0 Z 1 Z 2 2 α i j i j ≤ c1|z − z0| ∑ ∑ |∂ σ2 (z(t),ξ) − σ2 (z0,ξ) |dξdt α=2 0 i, j≤d 2 2 ≤ c2M |z − z0|

2 by the uniform L bound on the derivatives up to order 2 of σ2. This proves the lemma under this case. R 2 For (iii) first define the function h(z) = kσ2(z,ξ)−σ2(z0,ξ)k dξ. 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

2 2 h(z1) ≤ |∇zh(z0) · (z1 − z0)| + csupk∇z h(z)k|z1 − z0| z 2 ≤ C|z1 − z0| by the boundedness of the second derivatives of h (which follows from the assumptions on σ2). This gives us the result in this case. For case (iv), consider

σ2(µ1,z1,ξ) − σ2(µ2,z2,ξ) Z Z n n n = f (z1,x,ξ)d(µ1 − µ2 )(x) − ( f (z2,x,ξ) − f (z1,x,ξ))dµ2 (x) Rnd Rnd and hence through applications of Cauchy-Schwartz and Fubini, we get Z 2 kσ2(µ1,z1,ξ) − σ2(µ2,z2,ξ)k dξ Z 2 n n 2 ≤ 2 c(ξ) d(µ1 , µ2 ) dξ ZZ 2n 2 n + 2µ2(1) k f (z2,x,ξ) − f (z1,x,ξ)k dξdµ2 (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) 2 is L , and the deﬁnition of Lip1(nd) we get Z 2 n 2n 2 kσ2(µ1,z1,ξ) − σ2(µ2,z2,ξ)k dξ ≤ Cd(µ1, µ2) +C(µ1(1)µ2(1)) |z1 − z2|

2 2 ≤ L(µ1(1) ∨ µ2(1))[d(µ1, µ2) + |z1 − z2| ] 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 version of the martingale problem satisfied by the process constructed in [2] and an extension to the one considered in [13]. First we need to show that any solution (X,Z) d to (SE)Z0,K satisfies the following martingale problem (MP): X0 = µ ∈ MF (R ) and 2 d for any φ ∈ Cb(R ),

Z t X Mt (φ) ≡ Xt (φ) − µ(φ) − Xs(LXs φ)ds (2.19) 0 is a continuous martingale with square function

X hM (φ)it (2.20) Z t Z t ZZ 2 ∗ = Xs(φ )ds + ∇φ(z1)ρXs (z1,z2)∇φ (z2)dXs(z1)dXs(z2)ds 0 0

d where for ν ∈ MF (R ),

d d Z i 1 i j ∗ ∗ Lν φ(z) = ∑ b (ν,z)φi(z) + ∑ a (ν,z)φi j(z) with a = σ1σ1 + σ2σ2 (ξ)dξ i=1 2 i, j=1 (2.21) and ρν is the d × d matrix given by Z ∗ ρν (z1,z2) = σ2(ν,z1,ξ)σ2 (ν,z2,ξ)dξ. Rm R 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 d note that for φ : R → R bounded and measurable, (SE)Z0,K implies that Xt (φ) = R 2 d φ(Zt (y))Kt (dy). Then from Ito’sˆ Lemma (Theorem 2.2.11), for ψ ∈ Cb(R ) we

35 get

Z Z ψ(Zt )Kt (dy) = ψ(Z0)dK0(y) + Nt

Z t n ! 1 i j + Ks ∇ψ(Zs) · b(Xs,Zs) + ∑ ψi, j(Zs)a (Xs,Zs) ds, 0 2 i, j=1 where Nt is a martingale. This becomes, by the boundedness of ψ and its partials,

Z t Z Xt (ψ) = X0(ψ) + Nt + LXs ψ(z)Xs(dz)ds. 0 Since

Z t Z Z t Z Nt = ψ(Zs)dM(s,y) + Ks(∇ψ(Zs)σ2(Xs,Zs,ξ))dW(s,ξ), 0 0 by the orthogonality of M and W we have

Z t 2 hNit = Xs(ψ )ds 0 Z t Z Z Z ∗ + ∇ψσ2(Xs,z,ξ)dXs(z) ∇ψσ2(Xs,z,ξ)dXs(z) dξds, 0 which can be rewritten in the form of (2.20).

nd d Deﬁnition. For a function f : R → R, deﬁne Fn, f : MF (R ) → R by Fn, f (µ) = R f dµn.

Nn It can be shown, by induction and using Lemma 2.4.1, that for f = i=1 φi, 2 d where each φi ∈ Cb(R ), that

Z t Nt (n, f ) = Fn, f (Xt ) − Fn, f (X0) − LFˆ n, f (Xs)ds (2.22) 0

36 is a martingale, where

n ! ˆ LFn, f (µ) = ∑ µ(Lφi) ∏ µ(φ j) (2.23) i=1 j6=i ! 1 n Z + µ(φk) µ(φiφ j) + µ(∇φiσ2(ξ)) · µ(∇φ jσ2(ξ))dξ . ∑ ∏ m 2 i, j=1 k6=i, j R i6= j

We have suppressed the measure dependence of the generator L here and will do so below as well.

nd Deﬁnition. If f ∈ B(R ) (i.e. f a real valued, bounded Borel measurable function nd n nd (n−1)d on R ) then we can deﬁne the operator Φi, j : B(R ) → B(R ) acting on f as: n Φi, 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:

n Z n Z ˆ i n 1 n n−1 LFn, f (µ) = ∑ L f dµ + ∑ Φi, j( f )dµ (2.24) i=1 2 i, j=1 i6= j n d Z 2 1 k,l ∂ n + ∑ ∑ ρµ (zi,z j) f (z)dµ (z). 2 i, j=1 k,l=1 ∂zik ∂z jl i6= j

i Here z = (z1,...,zn) and L 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

Z t Z n ! Nt (n, f ) = ∑ ∏Xs(φ j) Xs(∇φiσ2(ξ))dW(s,ξ) (2.25) 0 i=1 j6=i Z t Z n ! + ∑ ∏Xs(φ j) φi(Zs)dM(s,y) 0 i=1 j6=i n Z t Z Z n = ∑ ∇zi f (z)σ2(zi,ξ)Xs (dz) dW(s,ξ) (2.26) i=1 0 n Z t Z Z n−1 i + ∑ f (Zs(y1),...,Zs(yn))Ks (dy ) dM(s,yi) i=1 0

i where y = (y1,...,yi−1,yi+1,...,yn). Then a calculation gives

n Z t Z ∗ 2n hN(n, f )it = ∑ ∇i f (z1)ρXs (zi,zn+ j)(∇n+ j f (z2)) Xs (dz) ds (2.27) i, j=1 0 n Z t Z Z 2n 2n−1 2n + ∑ Φi, j+n( f ⊗ f )Xs (dz ) ds i, j=1 0 where z = (z1,z2) with z1 = (z1,...,zn),z2 = (zn+1,...,z2n) and where 2n z = (z1,...,z2n−1).

2 nd Theorem 2.4.2. For any function f ∈ Cb(R ),

Z t Nt (n, f ) = Fn, f (Xt ) − Fn, f (X0) − LFˆ n, f (Xs)ds (2.28) 0 is a martingale, where Lˆ is given by (2.24).

Proof. Let

M nd O ∞ Hn = ∑ ai φi j : φi j ∈ Cc (R),ai ∈ R,M ∈ N . i=1 j=1

∞ nd We will ﬁrst use functions in Hn to prove that for f ∈ Cc (R ), (2.28) is a mar- 2 nd tingale and then boost that up to include functions in Cc (R ) and ﬁnally those in 2 nd ∞ nd Cb(R ). By the Stone-Weierstrass theorem, for any f ∈ Cc (R ), there is a se- α quence gk ∈ Hn such that gk → ∂ f uniformly, for α = (2,...,2). Then, by inte-

38 α α grating in the various components, we obtain a sequence fk such that ∂ fk → ∂ f uniformly for |α| ≤ 2. ˆ ˆ This in turn gives point-wise convergence of Fn, fk to Fn, f and LFn, fk to LFn, f d on MF (R ) (using the Dominated Convergence Theorem). Now let TN = inf{t : Kt (1) ≥ N} ∧ N as above. Then for each k,l, 2 P sup |Ns(n, fk) − Ns(n, fl)| ≤ cP(hN(n, fk) − N(n, fl)it∧TN ) s≤t∧TN and hence by sending l → ∞ and using the Dominated Convergence Theorem, we conclude that 2 P sup |Ns(n, fk) − Ns(n, f )| ≤ cP(hN(n, fk) − N(n, f )it∧TN ) s≤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

Z t 2n 2n−1 P(hN(n, f )it ) ≤ cP Xs(1) + Xs(1) ds . 0

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. 2 nd Now let f ∈ Cc (R ). Using approximate identities we can construct functions ∞ nd α α fk ∈ Cc (R ) such that ∂ fk → ∂ 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. 2 nd ∞ nd Finally, suppose f ∈ Cb(R ). Let ηk ∈ Cc (R ) such that ηk = 1 on B(0,k), c α ηk = 0 on B(0,k + 1) and k∂ ηkk < 2 for each multi-index with |α| ≤ 2. Letting α α fk = f ηk, we see that ∂ fk → ∂ f pointwise for each |α| ≤ 2. Here B(0,k) denotes the open ball of radius k about the origin. ˆ ˆ This gives LFn, fk → LFn, 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 martingale.

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 ( ,z) = (z) is Lipschitz.. Additionally, suppose that (z) ≡ R h( − σ1 µ σ1 η R ξ z)h(ξ)dξ is twice continuously differentiable with η0 and η00 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) :

n Z n Z ˆ 1 2 n 1 n LFn, f (µ) = ∑ (σ1 (zi)+η(0)) fii(z)dµ (z)+ ∑ ρ(zi,z j) fi j(z)dµ (z) 2 i=1 2 i, j=1 i6= j n Z 1 n n−1 + ∑ Φi, j( f )dµ 2 i, j=1 i6= j 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 d metric on MF (R ).

1 2 Theorem 2.5.1. Assume (Lip) holds. Let K ≤ K ≡ K be (Ft )-historical super- i d i d d Brownian motions with mi(·) ≡ E(K0(·)) ∈ MF (R ), and let Z0 : R → R be Borel maps, for i = 1,2. Let TN = inf{t : Kt (1) ≥ N} ∧ N. There are universal constants

40 i i {cN,N ∈ } so that if (X ,Z ) is the unique solution of (SE) i i , then N Z0,K

Z TN 1 2 2 P supd(X (r),X (r)) ds 0 r≤s Z 1 2 2 ≤ cN |Z0 − Z0 | ∧ 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 (X˜ i,Z˜i) = Φi(Xi,Zi) where each Φi depends on different historical super-Brownian motions i 1 2 dK1 K ,i = 1,2 and K ≤ K (ie, the Radon-Nikodym derivative dK2 ≤ 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 Ω = (Ω,F,(Ft )t≥0,Qν ). Let (X,Z) be the ˆ ∗ unique solution to (SE)Z0,K. Recall that X is (Ft )-predictable, and Z is (Ft )- d predictable. Denote by ΩX the space of continuous paths in MF (R ) and FX its Borel σ-field. t t Let ΩK = {H· ∈ C([0,∞),MF (C)) : Ht ∈ MF (C)∀t ≥ 0} where MF (C) = {µ ∈ t MF (C) : µ(A) = µ (A ∩ {y ∈ C : y = y })∀A ∈ C}. The historical super-Brownian motion K has sample paths in ΩK. Let FK be the Borel σ-field of ΩK and let

∞ K \ Ft = σ(Hr : 0 ≤ r ≤ t + 1/n). n=1

To treat W we will follow the method of [12]. Let QW be the law of the white d noise W. For functions f ∈ C(R ), deﬁne

−λ|x| k f kλ = sup | f (x)|e , x∈Rd

m and let Ctem = { f ∈ C(R ) : k f kλ < ∞ for all λ > 0}. We let Ctem be endowed with the topology generated by the family of norms k · kλ ,λ > 0. Hence, Ctem can ∞ −k be thought of as a metric space with metric d given by d( f ,g) = ∑k=1 2 (k f − 1 d gk1/k ∧ 1). It turns out that (Ctem,d) is a Polish space. Let W = (W ,...,W ) 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 d ΩW ≡ C([0,∞),Ctem). Endow ΩW with the topology of uniform convergence on compacts. d Let We : [0,∞) × ΩW → Ctem be the coordinate projection map, and let

∞ W \ Ft = σ(Wes : s ≤ t + 1/n). n=1

Then (H,We ) 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,We )(s) : s ≤ t) with the Qν,W -null sets thrown in. We will now treat H and We as random processes on the space ΩK,W with H(t,ω,α) = Ht (ω) and We (t,ω,α) = αt . Under Qν,W , H and We are independent by deﬁnition and have laws Qν and QW respectively. Recall from the deﬁnition in Section 2.2 that Ω[K,W = ΩK,W ×C and similarly ˆ ˆ ∗ Ht = Ht × Ct with universal completion Ht .

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 Ω. Then the following holds: ˆ ∗ ˜ (a) There are (Ht )-predictable and (Ht )-predictable maps X : R+ × ΩK,W → d d MF (R ) and Z˜ : R+ × Ω[K,W → R , respectively, depending only on (Z0,ν), and such that

X(t,ω),Z(t,ω,y) = X˜ t,K(ω),W(ω),Z˜t,K(ω),W(ω),y (2.29)

gives the unique solution of (SE)Z0,K.

(b) There is a continuous map from X 7→ 0 from M ( d) to M (Ω ) such that 0 PX0 F R 1 X ¯ if (X,Z) is a solution of (SE)Z0,K on some ﬁltered space Ω then Z (X ∈ ·) = 0 (·)d (ω). (2.30) P PX0(ω) P

42 (c) If T is an a.s. ﬁnite (Ft )-stopping time, then for all A ∈ FX

(X(T + ·) ∈ A|F )( ) = 0 (A), -a.s. P T ω PXT (ω) P

Proof. The proof of (a) is an analogue of the proof of Theorem V.4.1 (b) of [14]. ˜ ˜ For (a), let (X,Z) be the unique solution of (SE)Z0,K where (H,We ) is the canon- ¯ 0 ical process on Ω ≡ (ΩK,W ,H,Ht ,Qν,W ). It is clear from Theorem 2.3.3 that (X˜ ,Z˜) depends only on Z0 and ν, and satisﬁes the desired predictability conditions. To show (2.29), we need only show that the latter process solves (SE)Z0,K on ¯ 0 0 0 0 Ω. Let I ( f ,t,H,W,y),J (g,t,H,W,y) and f ∈ D1(d,d),g ∈ D2(m,d,d) denote the sample path stochastic integral and the white noise integral respectively from Sec- 0 tion 2.2 on Ω¯ . Similarly deﬁne I( f ,t),J(g,t), f ∈ D1,g ∈ D2 to be the stochastic ¯ 0 integrals in Section 2.2 on Ω. We claim that if f ∈ D1, then f ◦ (K,W)(t,ω,y) ≡ f (t,K(ω),W(ω),y) ∈ D1 and

I( f ◦ (K,W)) = I0( f ) ◦ (K,W),K-a.e. (2.31)

0 By the deﬁnition of D1 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 0 Proposition V.3.2 (d) of [14] allows us to show that it holds for all f ∈ D1. The 0 situation with the white noise integral J is similar in that if g ∈ D2, then J ◦(K,W) ∈ D2 and J(g ◦ (K,W)) = J0(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,We ) with (K(ω),W(ω)) in (SE)Z0,H and get that (X ◦(K,W),Z ◦(K,W)) solves ¯ (SE)Z0,K on Ω. 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,∞) × Ωˆ → (E,k · k) (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)k f (t,ω,y)k 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. ˆ n Theorem 2.5.3. Assume that f ∈ D1(n,d), g ∈ D2(m,n,d) and b : [0,∞)×Ω → R ˆ ∗ is (Ft )-predictable, and all three are locally bounded. Let Z(t) = Z0 + I( f ,t) + R t ˆ t J(g,t) + V(b,t) where V(b,t) = 0 b(s)ds. Deﬁne K(·) = Kt ({y : Z (ω,y) ∈ ·}). Then for a.a. ω for each k ∈ N there is a compact set Sk(ω) ⊂ C such that −1 supp(Kˆt ) ⊂ Sk ∀t ∈ [k ,k], and furthermore, supp(Kˆt ) ⊂ Sk ∀t ∈ [0,k] on {ω : supp(Kˆ0) 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 0 d for a.a. ω for each k ∈ N there is a compact set Sk(ω) ⊂ R such that supp(Xt ) ⊂ 0 −1 0 Sk ∀t ∈ [k ,k], and on {ω : supp(X0) is compact}, supp(Xt ) ⊂ Sk ∀t ∈ [0,k].

Proof. Note that under (Lip), the coefﬁcients σ1,σ2 and b are locally bounded and R −1 −1 that in the above, Xt (A) = 1(Z(t) ∈ A)Kt (dy) = Kˆt (πt A) where πt A = {y : Zt (ω,y) ∈ A} for A bounded measurable. Apply Theorem 2.5.3 and use the fact d that πt : C → R is a continuous mapping. −1 Let Sk be as in Theorem 2.5.3. Then for t ∈ [k ,k],

c ˆ −1 c Xt ((πt Sk) ) = Kt (πt ((πt Sk) )) ˆ c ≤ Kt (Sk) = 0.

0 Hence setting Sk = πkSk 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 Englander¨ [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:

Z t Z t n n−1 ¯ n n Yi (t) = Yp(i) (n − 1) + γ Yn(s) −Yi (s)ds + dBi (s), (3.1) n−1 n−1

n for n − 1 < t ≤ n, where Bi labels the ith particle of T alive from time n − 1 to n, p(i) is the parent of i and τn ¯ 1 n Yn(s) = ∑Yi (s) τn i=1 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 d values in the space of measures over the space of paths in R . 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

Bˆt = B·∧t is the underlying process of K. More information about K is provided in Section 3.2. d It can be shown that if a path y : [0,∞) → R 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 Z K Xt (·) = 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 Z 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(Y¯s −Ys)ds + dys Z 0 Xt (·) = 1(Yt (y) ∈ ·)Kt (dy) where the COM is deﬁned as

R xX0 dx ¯ s( ) Ys = R 0 . 1Xs(dx)

The details are given in Section 3.3 and it is shown that such a process X0 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, X0. It is shown in Theorem 3.4.1 that the COM converges in the case where X0 goes extinct in ﬁnite time, regardless of the value of γ. Theorem 3.4.2 shows that on the set where X0 survives indeﬁnitely, the COM

Y¯t converges in the attractive case (where γ > 0). Theorem 3.4.3 then shows that in 0 Xt the extinction case the mass normalized process 0 converges a.s. to a point as t Xt (1) 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 (·) a.s. −→ P∞ (·) as t → ∞ Xt (1) where P∞ (·) is the stationary distribution of the OU process, which is an extension of Proposition 20 of Englander¨ and Turaev [4] and Theorem 1 of Englander¨ and Winter [5] for the SOU process. The correspondence between X and X0 given in Section 3.3 is used in Theorem 3.4.11 to prove in the attractive case,

X0 (·) t a.s. Y¯∞ 0 −→ P∞ (·) as t → ∞, Xt (1)

Y¯∞ where the distribution P∞ (·) is equal to P∞ (·) shifted to the limiting value of the COM Y¯t . 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]. d t t Let C = C(R+,R ) with C its Borel σ-field and define C = {y : y ∈ C} where t s y = y·∧t and Ct = σ(y ,s ≤ t,y ∈ C). 0 0 0 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 E0 and the space of bounded

48 continuous functions from E to E0, 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 R n µ( f ) = f dµ. Furthermore, if f : E → R , deﬁne

µ( f ) = (µ( f1),··· , µ( fn)).

d For a point p ∈ R , let |p| denote Euclidean norm of p, and for a bounded function d d f : E → R , denote k f k = supx∈E ∑i=1 | fi(x)|.

t It turns out that Kt is supported on C ⊂ 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 Bˆt = B(·∧t) be the path-valued process associated with B, taking values in t s,y C . Then for φ ∈ bC, if s ≤ t let Ps,t φ(y) = E (φ(Bˆt )), where the right hand side denotes expectation at time t given that until time s, Bˆ follows the path y. The weak generator, Aˆ, of Bˆ is as follows. If φ : R+ ×C → R we say φ ∈ D(Aˆ) if and only if φ is bounded, continuous and (Ct )-predictable, and for some Aˆsφ(y) with the same properties as φ,

Z t φ(t,Bˆ) − φ(s,Bˆ) − Aˆrφ(Bˆ)dr,t ≥ s, s

s is a (Ct )-martingale under Ps,y for all s ≥ 0,y ∈ C . d If m ∈ MF R , we will say K satisﬁes the historical martingale problem, (HMP)m, if and only if K0 = m a.s. and

Z t Z t ∀φ ∈ D(Aˆ), Mt (φ) ≡ Kt (φt ) − K0(φ0) − Ks(Aˆsφ)ds − β Ks(φs)ds 0 0

(HMP)m is a continuous (Ft )-martingale with Z t 2 hM(φ)it = Ks(φs )ds ∀t ≥ 0, a.s. 0

Using the martingale problem (HMP)m, one can construct an orthogonal martingale 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

Z t 2 Ks(ψs )ds < ∞ ∀ t ≥ 0, (3.2) 0 then there exists a continuous local martingale Mt (ψ) with quadratic variation R t 2 hM(ψ)it = 0 Ks(ψs )ds. If the expectation of the term in (3.2) is ﬁnite, then Mt (ψ) is an L2-martingale. ˆ ˆ ˆ ˆ ∗ Deﬁnition. Let (Ω,F,Ft ) = (Ω ×C,F × C,Ft × Ct ). Let Ft denote the universal completion of Fˆ t . If T is a bounded (Ft )-stopping time, then the normalized ˆ ˆ ˆ Campbell measure associated with T is the measure PT on (Ω,F) given by

ˆ P(1AKT (B)) PT (A × B) = for A ∈ F,B ∈ C. mT (1)

ˆ where mT (1) = P(KT (1)). We denote sample points in Ω by (ω,y). Therefore, un- ˆ −1 der PT , ω has law KT (1)dP·mT (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 X0. The symbol X˚ will be used as a catch-all and will vary according to the setting, but will be a measure- d valued process taking values in MF R . Occasionally the symbol K˚ will appear and will denote a historical process different from K.

3.3 Existence and preliminary results ˆ ∗ 1 2 1 Deﬁnition. For two (Ft )-measurable processes X and X , we will say that X = X2,K-a.e. if 1 2 X (s,ω,y) = X (s,ω,y),∀s ≤ t,Kt -a.a. y for all ﬁxed times t ≥ 0.

(X,Z) 1 We say that is a solution to the strong equation (SE)Z0,K if it satisﬁes

Z t 1 Z ( ,y) = Z ( ,y ) + y − y − Z ( ,y)ds K − (SE)Z0,K (a) t ω 0 ω 0 t 0 γ s ω a.e. 0 Z d (b) Xt (A) = 1(Zt ∈ A)Kt (dy) ∀A ∈ B(R ),∀t ≥ 0,

50 ˆ ∗ where Zt is an (Ft )-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. 1 2 Note that in the deﬁnitions of (SE)Z0,K and (SE)Y0,K given below, part (b) is unnecessary to solve the equations. It has been included to provide an easy comparison to the strong equation of chapter V.1 of [14].

There is a pathwise unique solution (X,Z) to (SE)1 . That is, X Theorem 3.3.1. Z0,K 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.

−βt −βt Proof. Note that K˚t = e Kt deﬁnes a (∆/2,0,e )-Historical superprocess. Let ˆ 1 ˚ PT be the Campbell measure associated with K (note that if T is taken to be a ˆ random stopping time, this measure differs from PT ). The proof of Theorem V.4.1 of [14] with minor modiﬁcations shows that (SE)1 has a pathwise unique so- Z0,K˚ ˆ 1 lution. This is because (K3) of Theorem 2.6 of [13] shows that under PT , 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 (X˚ ,Z) is the solution of (SE)1 , and we let Z0,K˚ Z βt Xt (·) ≡ e X˚t (·) = 1(Zt ∈ ·)Kt (dy)

(X,Z) 1 then is the pathwise unique solution of (SE)Z0,K. The only thing to check R t this is that Zt (ω,y) = Z0(ω,y0) + yt − y0 − γ 0 Zs(ω,y)ds K-a.e., but this follows from the fact that K˚t Kt ,∀t. It can be shown by using by Theorem 2.14 of [13] that X satisﬁes the following 2 d martingale problem: For φ ∈ Cb(R ),

Z t Z ∆ Z t Mt (φ) ≡ Xt (φ) − X0(φ) − −γx · ∇φ(x) + φ(x)Xs(dx)ds − β Xs(φ)ds 0 2 0

51 R t 2 is a martingale where hM(φ)it = 0 Xs(φ )ds. Then by Theorem II.5.1 of [14] this implies that X is a version of a super Ornstein-Uhlenbeck process, with initial −1 distribution given by K0 Z0 (·) .

Remark 3.3.2. (a) Under the Lipschitz assumptions of Section V.1 of 1 K [14], one can in fact uniquely solve (SE)Z0,K where is a supercritical historical Brownian motion. The proof above can be extended with minor mod- iﬁcations.

ˆ (b) The proof of Theorem 3.3.1 essentially shows that under PT , T ﬁxed, the + ˆ d path process y : R × Ω → R such that (t,(ω,y)) 7→ yt is a d-dimensional Brownian motion stopped at T.

ˆ (c) Under PT , T ﬁxed, Zt can be written explicitly as a function of the driving path: For t ≤ T,

Z t Z t γt γs γs e Zt = Z0 + e dZs + Zs (γe )ds (3.3) 0 0 Z t Z t Z t γs γs γ = Z0 + e dys + e (−γZs)ds + Zs(γe s)ds. 0 0 0 Hence, Z t −γt −γ(t−s) Zt (y) = e Z0 + e dys. 0 (X0,Y) 2 We say is the solution to the stochastic equation (SE)Y0,K if:

Z t 2 (a) Y ( ,y) = Y ( ,y) + y − y + Y¯ ( ) −Y ( ,y)ds,K y (SE)Y0,K t ω 0 ω t 0 γ s ω s ω -a.a. 0 Z 0 d (b) Xt (A) = 1(Yt ∈ A)Kt (dy),∀A ∈ B(R ),∀t ≥ 0.

ˆ ∗ 0 where Yt is an (Ft )-predictable process, Xt is an (Ft )-predictable process and

R xX0(dx) K (Y ) ¯ t t t Yt ≡ 0 = , Xt (1) Kt (1) is the center of mass of X0. Intuitively, X0 is an interactive measure-valued diffusion where the representative 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 (X0,Y) can be naturally reformulated as questions about the ordinary SOU process.

(X,Z) 1 Notation. Unless stated otherwise, will refer to a solution of (SE)Z0,K and (X0,Y) 2 to the solution of (SE)Y0,K. ˆ ∗ d Deﬁnition. For an arbitrary Ft -adapted, R -valued process zt , deﬁne the centre of mass (COM),z ¯t , with respect to Kt as follows:

Kt (zt ) z¯t = . Kt (1)

We also let

X K Z X0 K Y K ˜ t (·) t ( t ∈ ·) ˜ 0 t (·) t ( t ∈ ·) ˜ t (·) Xt (·) ≡ = , Xt (·) ≡ 0 = and Kt (·) = . Xt (1) Kt (1) Xt (1) Kt (1) 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 ofz ¯t for t < η where η is the extinction time. 2 Next we show that there exists a solution to (SE)Y0,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 K ( ) Y¯ 1 factor t 1 in the definition of t . Instead we will utilize the solution of (SE)Y0,K 2 to define a unique solution for (SE)Y0,K.

There is a pathwise unique solution to (SE)2 . Theorem 3.3.3. Y0,K

Proof. Y 2 ˆ Y Suppose there exists a solution satisfying (SE)Y0,K. Then under PT , t can be written as a function of the driving path y and Y¯ . Using integration by parts gives

Z t Z t γt γs γs e Yt = Y0 + e dYs + γe Ysds 0 0 Z t Z t γs γs = Y0 + e dys + γe Y¯sds 0 0

53 and hence Z t Z t −γt γ(s−t) γ(s−t) Yt = e Y0 + e dys + γe Y¯sds. (3.4) 0 0 (X,Z) 1 Z = Y If is the solution to (SE)Z0,K where 0 0, then note that by Remark 3.3.2 (c), Z t γ(s−t) Yt = Zt + γe Y¯sds. 0

By taking the normalized measure K˜t of both sides of the above equation, we get

Z t −γ(t−s) Y¯t = Z¯t + γ e Y¯sds (3.5) 0

Hence Y¯t 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 Z t Y¯t = Z¯t + γ Z¯sds, 0 ¯ 1 which is easily veriﬁed using integration by parts. Also, if Yt is a second process which solves (3.5), then

Z t ¯ ¯ 1 −γ(t−s) ¯ ¯ 1 |Yt −Yt | = γ e (Ys −Ys )ds 0 Z t −γ(t−s) ¯ ¯ 1 ≤ |γ| e |Ys −Ys |ds. 0

¯ ¯ 1 By Gronwall’s inequality, this implies Yt =Yt , for all t and ω. Pathwise uniqueness X0 1 of follows from the uniqueness of the solution to (SE)Z0,K and the uniqueness of the process Y¯t solving (3.5). 2 We have shown that if there exists a solution to (SE)Y0,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 X0 as functions of the pathwise unique solution to

54 1 Z = Y (SE)Z0,K where 0 0:

Z t Yt = Zt + γ Z¯sds 0 0 Xt (·) = Kt (Yt ∈ ·).

Then Y¯t satisﬁes the integral equation (3.5), and hence

Z t Z t −γ(t−s) Z¯sds = e Y¯sds. 0 0 Therefore

Z t −γ(t−s) Yt = Zt + γ e Y¯sds 0 Z t Z t −γt −γ(t−s) −γ(t−s) = e Y0 + e dys + γ e Y¯sds, 0 0 and so Z t Z t γt γs γs e Yt = Y0 + e dys + γ e Y¯sds. 0 0 Multiplying by e−γt and using integration by parts shows

Z t Yt = Y0 + yt − y0 + γ (Y¯s −Ys)ds 0

K y (X0,Y) 2 which holds for -a.e. , thereby showing satisﬁes (SE)Y0,K.

Remark 3.3.4. Some useful equivalences in the above proof are collected below.

If Y0 = Z0, then for t < η,

Z t (a) Yt = Zt + γ Z¯sds 0 Z t (b) Y¯t = Z¯t + γ Z¯sds 0

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.

0 d We now consider the martingale problem for X . For φ : R → R, let φ¯t ≡ Kt (φ(Yt ))/Kt (1). Note that the lifetime of the process φ¯ is [0,η). Then the following theorem holds:

2 d Theorem 3.3.5. For φ ∈ Cb(R ,R), and t < η,

Z t φ¯t = φ¯0 + Mt + b¯sds 0 where 1 b = γ∇φ(Y ) · (Y¯ −Y ) + ∆φ(Y ) s s s s 2 s and Mt is a continuous local martingale on its lifetime such that

t Z Z φ(Ys) − φ¯s Mt = dM(s,y) 0 Ks(1) and hence has quadratic variation given by

Z t 2 ¯ 2 φs − (φs) [M]t = ds. 0 Ks(1)

Proof. The proof is not very difficult; one need only use Ito’sˆ 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. ˆ Let T be a fixed time and t ≤ T. Recall that under PT , y is a stopped Brownian motion by Remark 3.3.2 (b), and hence Yt (y) is a stopped Ornstein-Uhlenbeck

56 ˆ process (attracting to Y¯t ). Hence, by the classical Ito’sˆ Lemma we have, under PT ,

Z t Z t 1 i j φ(Yt ) = φ(Y0) + ∇φ(Ys) · dYs + ∑ φi j(Ys)d[Y ,Y ]s 0 2 i, j≤d 0 Z t Z t 1 = φ(Y0) + ∇φ(Ys) · dys + γ∇φ(Ys) · (Y¯s −Ys) + ∆φ(Ys)ds. 0 0 2 Then

Z t Z t Kt (φ(Yt )) = Kt (φ(Y0)) + Kt ∇φ(Ys) · dys + Kt bsds 0 0 Z t Z Z t = K0(φ(Y0)) + φ(Y0)dM(s,y) + β Ks (φ(Y0))ds 0 0 Z t Z Z s Z t Z s + ∇φ(Yr) · dyr dM(s,y) + β Ks ∇φ(Yr) · dyr ds 0 0 0 0 Z t Z Z t Z t + bsdM(s,y) + β Ks(bs)ds + Ks(bs)ds 0 0 0 Z t Z = K0(φ(Y0)) + φ(Ys)dM(s,y) 0 Z t Z t + β Ks (φ(Ys))ds + Ks(bs)ds. 0 0

R t The equality in the third line to Kt 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 Y¯ .

Supposing t < η, Ito’sˆ formula implies (with properties of Kt (1) and Kt (φ(Yt )))

Kt (φ(Yt )) φ¯t = Kt (1) Z t Z Z t ¯ φ(Ys) Ks(φ(Ys)) Ks(bs) = φ0 + − 2 dM(s,y) + ds 0 Ks(1) Ks(1) 0 Ks(1)

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 martingale 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 2 d β-supercritical SOU process, X, for φ ∈ Cb(R ,R) and t < η,

Z t X˜t (φ) = K˜t (φ(Zt )) = K˜0(φ(Z0)) + Nt + K˜s(Lφ(Zs))ds 0 where 1 Lφ(x) = −γx · ∇φ(x) + ∆φ(x) 2 and Nt is a continuous local martingale on its lifetime such that

t Z Z φ(Zs) − K˜s(φ(Zs)) Nt = dM(s,y) 0 Ks(1) and hence has quadratic variation given by

t 2 2 Z K˜s φ (Zs) − K˜s(φ(Zs)) [N]t = ds. 0 Ks(1)

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

−βt e Xt (1) → W a.s. and {η < ∞} = {W = 0} almost surely.

−βt −βt Proof. Note that X˚t = e Xt is a (∆/2,0,e )-superprocess. The martingale problem then shows that X˚t (1) is a non-negative (Ft )-martingale and therefore converges 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

−2βX (1) P(η < ∞) = e 0 .

The same exercise also shows ! X˚ ˚ 2λβ 0(1) P0,X0 (exp(−λXt (1))) = exp − . 2β + λ 1 − e−βt

58 Now sending t → ∞ gives

˚ ! −λW 2βλX0(1) P0,X0 (e ) = exp − , 2β + λ and sending λ → ∞ gives

−2βX˚ (1) P(W = 0) = e 0 = P(η < ∞) since X0 = X˚0. Therefore {η < ∞} = {W = 0} almost surely.

+ 1/2 + Deﬁne h(δ) = (δ ln (1/δ)) 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 α be chosen large enough so that pc0 (λ) = C(d,c0)λ 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 n measure m under Qτ,m and A a Borel subset of C(R ), then the process deﬁned by

0 Ht (·) = Ht (· ∩ {y : yτ ∈ A}) is also a supercritical historical Brownian motion starting at time τ with initial 0 0 measure m given by m (·) = m(· ∩ A) under Qτ,m. Then using the extinction probabilities for H0, we have

2βm(A) Qτ,m (Ht ({y : yτ ∈ A}) = 0 ∀t ≥ s) = exp − . (3.6) 1 − e−β(s−τ)

59 j Using the Markov property for K at time 2n and (3.6) gives j + 1 j j − 1 j −n P ∃t > s.t. Kt y : y − y > c h 2 > 0 2n 2n 2n 2n (3.7)

  n j j−1 j −n o 2βK j y : y n − y n > c n h(2 )  n 2 2 2  = 1 − exp − 2 P −β 1   1 − e 2n 

 n j j−1 j −n o 2βK j y : y n − y n > c n h(2 ) n 2 2 2 ≤ 2 P −β 1  1 − e 2n

 n j j−1 j −n o 2βK j y : y n − y n > c n h(2 ) n 2 2 2 ≤ 2 P β β 2  2n − 22n+1 n+1 2 ˆ j j − 1 j −n ≤ P K j (1) P j y n − y n > c n h 2 . β 2n 2n 2 2 2 1 − 2n+1

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:

n+1 j 2 2 d/2−1 −nc( n ) /2 (3.7) ≤ P K j (1) cdn 2 2 β 2n 1 − 2n+1 β j ln2 j n+2 n d/2−1 −n( n +c0)/2 ≤ 2 2 2 P(X0(1))cdn 2 2 .

60 Hence, summing over j from 1 to n2n gives

j + 1 There exists 1 ≤ j ≤ n2n s.t. ∃t > s.t. P 2n j j − 1 j −n Kt y : y − y > c h 2 > 0 2n 2n 2n n n 2 β j ln2 j d/2−1 n+2−nc0/2 n −n( n ) ≤ P(X0(1))cdn 2 ∑ 2 2 2 j=1 n n 2 j d/2−1 −2n+2 − n ≤ P(X0(1))cdn 2 ∑ 2 2 j=1 d/2−1 −n+2 ≤ P(X0(1))cdn 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, n j+1 for all 1 ≤ j ≤ n2 , for all t ≥ 2n , for Kt -a.a. y, j − 1 j j −n y − y < c h 2 . 2n 2n 2n

Letting δ(ω) = 2−N(ω), note that on the dyadics, by above, we have that

lnλ (δ < λ) = N > − P P ln2 lnλ j + 1 = ∃n > − ,∃ j ≤ n2n,∃t > , s.t. P ln2 2n j j − 1 j −n Kt y : y − y > c h 2 > 0 2n 2n 2n

0 d/2−1 2n−nc0/2 ≤ ∑ C (d,c0)n 2 lnλ n=b− ln2 c c 0 −ε ≤ C(d,c0,ε)λ 2 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

Y¯t . Recall that η is the extinction time of K.

2 2 Lemma 3.3.9. Assume P K˜0 |Z0| + |y0| < ∞. Then,

2 P |Yt | ;t < η < A(γ,t), where   O1 +t6e−2γt if γ < 0 A(γ,t) =  O1 +t5 if γ ≥ 0.

Proof. Assume that t < η. Recall

Z t −γt γ(s−t) Zt (ω,y) ≡ e Z0 + e dys. 0

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 R t ¯ lemma. Then, as Yt = Zt + γ 0 Zsds,

Z t 2 2 2 2 |Yt | . |Zt | + γ t |Z¯s| ds 0 Z t 2 2 2 . |Zt | + γ t |Zs| ds 0 by Cauchy-Schwartz and Jensen’s inequality. Therefore integrating with respect to the normalized measure gives

Z t 2 2 2 2 |Yt | ≤ |Zt | + γ t |Zs| ds (3.8) 0

2 and therefore we need only ﬁnd the appropriate bounds for expectation of |Zt | to get the result.

62 After another few applications of Cauchy-Schwartz and integrating by parts,

Z t 2 2 −2γt 2 −γt γs |Zt | . e |Z0| + e e dys 0 Z t 2 −2γt 2 −2γt γt γs . e |Z0| + e e yt − y0 − γ yse ds 0 Z t −2γt 2 2 2 2 2 −2γ(t−s) . e |Z0| + |y0| + |yt | + γ t |ys| e ds. 0

As y ∈ S(δ,c(t)),

tc(t)h(δ)2 |Z |2 e−2γt |Z |2 + |y |2 + |y |2 + t . 0 0 0 δ Z t " 2# 2 2 sc(s)h(δ) −2γ(t−s) + γ t |y0| + e ds 0 δ −2γt 2 2 2 2 −2γt . e |Z0| + |y0| + |y0| + |y0| γt 1 − e /2 h(δ)2 + c(t)2 t2 + γt3 1 − e−2γt δ −2γt 2 2 . (1 + |γ|t) 1 + e |Z0| + |y0| h(δ)2 + c(t)2 t2 + γt3 1 − e−2γt δ

Integrating by the normalized measure K˜t ,

2 −2γt ˜ 2 2 |Zt | . (1 + |γ|t) 1 + e Kt |Z0| + |y0| h(δ)2 + c(t)2 t2 + γt3 1 − e−2γt . δ

2 Then using (3.8) and using the above bound on |Zt | gives

Z t 2 2 2 2 |Yt | ≤ |Zt | + γ t |Zs| ds 0 Z t 2 −2γt ˜ 2 2 ˜ 2 2 . (1 +t ) 1 + e Kt |Z0| + |y0| + Ks |Z0| + |y0| ds (3.9) 0 h(δ)2 + (1 +t)c(t)2 t2 + γt3 1 − e−2γt . δ

63 2 2 Note that φ(y) ≡ |Z0(y)| + |y0| and φn(y) ≡ φ(y)1(|y| ≤ n) are Fˆ 0 measurable. −1 By applying Ito’sˆ formula to Kt (φn)Kt (1) and using the decomposition Kt (φn) = R t R t K0(φn) + 0 φn(y)dM(s,y) + β 0 Ks(φn)ds (which follows from Proposition 2.7 of [13]) we get

K˜t (φn) = K˜0(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,

K˜t (φ);t < η = lim K˜t (φn);t < η P n→∞ P = lim lim P K˜t (φn)1(t < TN) n→∞ N→∞ ˜ ˜ = lim lim P Kt∧TN (φn) − KTN (φn)1(t ≥ TN) n→∞ N→∞ ˜ ≤ lim lim P Kt∧TN (φn) n→∞ N→∞ ˜ = lim lim P K0(φn) + Nt∧TN (φn) n→∞ N→∞ = lim K˜0(φn) n→∞ P = P K˜0(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

Z t Z t P K˜s(φ)ds;t < η ≤ P K˜s(φ)1(s < η)ds 0 0 Z t = P K˜s(φ);s < η ds 0 ≤ tP K˜0(φ) , by the calculation immediately above. Therefore, taking expectations in (3.9) and

64 √ plugging in c(t) = c0 +t gives

2 2 3 −2γt 5 −2γt h(δ) P |Yt | ;t < η 1 +t 1 + e P K˜0 (φ) +t 1 +te P . . δ 2

α Now let c0 be chosen so that supp(Kt ) ⊂ S(δ,c(t)) and pc0 (λ) =Cλ for λ ∈ [0,1]. Note that

h(δ)2 ln+(1/δ) Z ∞ ln+(1/λ) P 2 = P = dP(δ < λ) δ δ 0 λ Z 1 ln+(1/λ) ≤ dP(δ < λ) + P(δ > 1) 0 λ ln+(1/λ) = lim P(δ < λ) − P(δ < 1) λ↓0 λ Z 1 ln+(1/λ) − P(δ < λ)d + P(δ > 1) 0 λ < ∞, 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

2 P |Zt | ;t < η < A(γ,t).

(b) Lemma 3.3.9 and its proof can be extended to show that for any positive in- k k teger k if P K˜0 |Z0| + |y0| < ∞, then there exists a function B(γ,t,k) polynomial in t if γ ≥ 0, exponential if γ < 0 such that

k k P |Yt | ;t < η < B(γ,t,k) and P |Zt | ;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 " # h(δ)k P < ∞, δ

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 0 R R R 0 X and X , φ(y0)K0(dy) = φ(x)X0(dx) = φ(x)X0(dx). 0 Assume that X0(1) = X0(1) < ∞ for Theorems 3.4.1, 3.4.2 and 3.4.3 below. c Theorem 3.4.1. On S , Y¯t and Z¯t converge as t ↑ η < ∞, P-a.s., for any γ ∈ R. 2 Proof. Assume for now that P K˜0 |y0| < ∞ (the case where K0 = 0 can be ignored without loss of generality). By Theorem 3.3.5,

t Z Z Ys −Y¯s Y¯t = Y¯0 + dM(s,y) 0 Ks(1) 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, sup |Y¯s| > n = lim sup |Y¯s| > n P N→ P s

By the ﬁrst Borel-Cantelli lemma, P sup |Y¯t | > n i.o. = 0. s

It follows that

liminfY¯s > −∞ and limsupY¯s < ∞ s→t∧η s→t∧η

66 which implies that on the set {η < t}, Y¯s converges, by Theorem IV.34.12 of Rogers c and Williams [18]. This shows convergence on the extinction set as S = ∪t {η < t}. Note that if ν(·) = P(K0 ∈ ·). Theorem II.8.3 of [14] gives Z P(K ∈ ·) = PK0 (K ∈ ·)dν(K0),

where PK0 is the law of a historical Brownian motion with initial distribution δK0 . 2 Hence, the a.s. convergence of Y¯t in the case where K˜0 |y0| is ﬁnite in mean imply

Z ¯ c ¯ c P limYt exists;S = PK0 limYt exists;S dν(K0) (3.10) t↑η t↑η Z c = PK0 (S )dν(K0)

c = P(S )

2 if K0 |y0| + 1 < ∞,ν-a.s. 2 Finally, to get rid of the assumption that K0 |y0| < ∞ note that Corollary 3.4 0 of [13] ensures that if K0(1) < ∞, then at any time t > 0, Kt (and hence Xt ,Xt ) is compactly supported. Therefore, letting Sr = {Kr 6= 0} we see that

! ¯ c [ ¯ c P limYt does not exist,S = P limYt does not exist,S1/r,S t↑η t↑η r∈N ¯ c ≤ P PK /r limYt does not exist,S 1 S1/r ∑ 1 t↑η r∈N = 0

2 by 3.10 since K1/r a.s. compact implies that K1/r |y1/r| < ∞ holds. This completes the proof for the convergence of Y¯ on Sc in its full generality.

The convergence of Z¯t now follows from the convergence of Y¯t and Equa- tion 3.5.

Theorem 3.4.2. On S the following hold:

67 (a) If γ > 0 ∞ a.s. a.s. Z Z¯t −→ 0 and Y¯t −→ γ Z¯sds, 0 and this integral is ﬁnite almost surely.

(b) If γ = 0, then Z¯t = Y¯t converges almost surely.

2 Proof. Let γ ≥ 0 and as in the previous theorem assume P K˜0 |y0| < ∞. Also, without loss of generality, assume that d = 1 for this proof. By Theorem 3.3.5, Y¯t is a continuous local martingale with decomposition given by Y¯t = Y¯0 + Mt (Y) where

Z t 2 ¯ 2 Z t Ys −Ys V(Ys) [M(Y)]t = ds ≡ ds. 0 Ks(1) 0 Ks(1)

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 = −βt limt→∞ e Kt (1). Hence it follows that [M(Y)]∞ < ∞ on S if

Z ∞ −βs e V(Ys)ds < ∞. 0 Then ! Z ∞ V(Y ) Z ∞ Y 2 s ds S ≤ s ds S P s ; P s ; 0 eβ 0 eβ Z ∞ A(γ,s) ≤ ds s 0 eβ < ∞ by Cauchy-Schwartz and Lemma 3.3.9 since γ ≥ 0. Therefore on S, Y¯t converges a.s. to some limit Y¯∞. Note that if γ = 0, Remark 3.3.4 (b) gives Y¯t = Z¯t and so (b) holds.

That Z¯t converges on S for γ > 0 follows from the fact that Y¯t converges and

68 Equation 3.5 by setting

Z t −γ(t−s) Z¯t = Y¯t − γ e Y¯sds 0 Z t −γ(t−s) −γt = Y¯t −Y¯∞ + γ e (Y¯∞ −Y¯s)ds + e Y¯∞ 0 → 0 as t → ∞.

¯ ¯ R t ¯ ¯ a.s. By Remark 3.3.4 (b) we see that for γ > 0 since Yt = Zt + γ 0 Zsds, Zt −→ 0 ¯ a.s. R ∞ ¯ and Yt −→ γ 0 Zsds. 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,

˜ ˜ 0 Xt → δF and Xt → δF0

0 d as t ↑ η < ∞ a.s., where F and F are R -valued random variables such that

Z η 0 F = F + γ Z¯sds. 0 Proof. As in the previous theorems, note that we need only consider the case that 2 P K˜0 |y0| < ∞. We will follow the proof of Theorem 1 of Tribe [20] here. Deﬁne

Z t 1 ζ(t) = ds,t < η. 0 Ks(1)

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,

Z t √ zt = z0 + zsdBs,Q-a.s. 0 Then using Girsanov’s Theorem, (Theorem V.27.1 of [18]) we see that there is a

69 probability measure 0 such that, dQ | = N where Q dQ0 Ft t

Z t β 2 Z t Nt = exp βdzs − zsds 0 2 0

0 is an exponential martingale. Furthermore, under Q , zt solves

Z t Z t √ 0 zt = z0 + zsdBs + βzsds,Q -a.s. 0 0

c Let St = {ω : ∃s ≤ t,zs = 0}. Then

Z Z η Z Z η 1 0 1 1 ds < ∞ d = 1 ds < ∞ Nt d = 0, c Q c Q St 0 zs St 0 zs

c R η −1 0 by the result of Konno and Shiga. Note then that on ∪St , 0 zs ds = ∞, Q -a.s. Since Kt (1) is a diffusion that satisfies the previous SDE with respect to P, ζ satis- c fies the required properties on S , 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

XD XD = X0 ,X˜ D = t G = F t Dt t D and t Dt Xt (1) and deﬁne 1 L φ(x) = γ(Y¯ − x) · ∇φ(x) + ∆φ(x). t t 2 Let Z ηN 1 TN = ds, 0 Ks(1) where ηN = inf{s : Ks(1) ≤ 1/N}. Then note that TN ↑ ζ(η−) and each TN is a

70 c 2 Gt -stopping time. On S for φ ∈ Cb, Theorem 3.3.5 implies,

Z Dt∧T ˜ D ˜ N ˜ 0 Xt∧TN (φ) = X0(φ) + Xs(Lsφ)ds + MDt∧T (φ) 0 N Z t∧TN ˜ ˜ D D = X0(φ) + Xs (LDs φ)Xs (1)ds + Nt∧TN (φ) 0 Z t∧TN ˜ D = X0(φ) + Xs (LDs φ)ds + Nt∧TN (φ) (3.11) 0

D since dDt = Xt (1)dt and where Nt = MDt . It follows that Nt∧TN is a Gt -local martingale. Then, by Theorem 3.3.5,

D 0 2 0 2 Z t∧TN X˜ ( ) − X˜ ( ) ˜ D s φ s φ X (φ) t∧T = ds N 0 Xs(1) Z t∧TN ˜ D 2 ˜ D 2 = Xs (φ ) − Xs (φ) ds, 0 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,

Z ∞ Z ∞ Z D ¯ 1 D Xs (|LDs φ|)ds ≤ |γ(YDs − x) · ∇φ(x)| + |∆φ(x)|Xs (dx)ds 0 0 2 Z ∞ ¯ 1 D ≤ |γ|k∇φkKDs (|YDs −YDs |) + k∆φkXs (1)ds 0 2 Z ∞ D ˜ ¯ 1 = Xs (1) |γ|k∇φkKDs (|YDs −YDs |) + k∆φk ds 0 2 Z ∞ 1 D 2 2 1 ≤ Xs (1) |γ|k∇φk |YDs | + k∆φk ds 0 2 where in the second line we have used the deﬁnition of X0 and the Cauchy-Schwartz

71 inequality in the fourth. Using the deﬁnition of Ds yields

Z ∞ 1 Z ∞ D 2 2 1 D Xs (|LDs φ|)ds ≤ |γ|k∇φksup |Ys| + k∆φk Xs (1)ds 0 s<η 2 0 1 2 2 1 = |γ|k∇φksup |Ys| + k∆φk η s<η 2 < ∞ (3.12)

2 2 as φ ∈ Cb and |Ys| is continuous on [0,η) (which follows from Theorem 3.3.5). Hence, this implies that for φ positive, on Sc

Z ∞ ˜ D Nt (φ) > −X0(φ) − Xs (|LDs φ|)ds 0 for all t and hence by Corollary IV.34.13 of [18], Nt converges as t → ∞. Therefore ˜ D by (3.11) and (3.12), Xt (φ) converges a.s. as well. ˜ D ˜ D ˜ D Denote by X∞ (φ) the limit of Xt (φ). It is immediately evident that X∞ (·) is a d ˜ D 0 probability measure on R . To show that X∞ (·) = δF0 where F is a random point d in R , we now defer to the proof of Theorem 1 in Tribe [20], as it is identical from this point forwards. c Similar (but simpler) reasoning holds to show X˜t → δF a.s. on S where F is a d R t ¯ random point in R . Let f (t) = γ 0 Zsds. Note that f is independent of y and that f (t) → f (η) a.s. when t ↑ η because Y¯t = Z¯t + f (t) and both Y¯t and Z¯t converge a.s. by Theorem 3.4.1. Then for φ bounded and Lipschitz,

Z Z 0 0 φ(x − f (t))X˜ (dx) − φ(x − f (η))X˜ (dx) ≤ C | f (η) − f (t)| t t −→a.s. 0. as t ↑ η. Therefore it is enough to note that since f (η) depends only on ω, the ˜ 0 convergence of Xt gives Z ˜ 0 a.s. 0 φ(x − f (η))Xt (dx) −→ φ F − f (η)

72 and hence Z ˜ 0 a.s. 0 φ(x − f (t))Xt (dx) −→ φ F − f (η) .

By Remark 3.3.4 (a), Z Z ˜ 0 ˜ φ(x − f (t))Xt (dx) = φ (Yt (y) − f (t))Kt (dy) Z = φ (Zt (y))K˜t (dy) Z = φ(x)X˜t (dx)

−→a.s. φ (F) as t ↑ η. Since there exists a countable separating set of bounded Lipschitz functions {φn}, and the above holds for each φn,

Z η 0 F = F + γ Z¯sds, 0 a.s.

Remark 3.4.4. (a) Theorem 3.4.3 holds in the critical branching case. That is, if β = 0, a.s. 0 a.s. Xt −→ δF and Xt −→ δF0

where Z η 0 F = F + Z¯sds. 0 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 X˜0 and η is the extinction time. Finding the distribution of F0 remains an open problem however.

The next few results are necessary to establish the almost sure convergence of

73 X˜t on the survival set. This will in turn be used to show the almost sure convergence ˜ 0 of Xt using the correspondence of Remark 3.3.4 (a). d Deﬁne Lip1 = ψ ∈ C(R ) : ∀x,y,|ψ(x) − ψ(y)| ≤ |x − y|,kψk ≤ 1 . Deﬁne Pt as the standard Ornstein-Uhlenbeck semigroup (with attraction to the origin).

Note that Pt → P∞ in norm where

d Z γ 2 2 P φ(x) = φ(z) e−γ|z| dz, ∞ π

−βt which is independent of x. Recall that W = limt→∞ e Xt (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 problem for X. Let A be the inﬁnitesimal generator for an Ornstein-Uhlenbeck process, d and hence recall that for φ ∈ C2(R ),

∆ Aφ(x) = −γx · ∇φ(x) + φ(x). 2

4 4 Lemma 3.4.5. If γ > 0, P X0 |x| < ∞ and P X0(1) < ∞, then on S, for any 2 −βt L φ ∈ Lip1, e Xt (φ) −→ WP∞φ and

2 −βt −ζt P e Xt (φ) −WP∞φ ≤ Ce 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 d Proposition II.5.7 of [14], for functions ψ : [0,T] × R → R such that ψ satisﬁes the deﬁnition before that Proposition,

Z t Z Z t Xt (ψt ) = X0(ψ0) + ψs(x)dM(x,s) + Xs (Aψs + βψs + ψ˙s)ds 0 0 where M is the orthogonal martingale measure derived from the martingale problem 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

Z t Z Z t Xt (φ) = X0(Pt φ) + Pt−sφ(x)dM(s,x) + βXs(Pt−sφ)ds 0 0

∂ −βt since ∂s Psφ = APsφ. Multiplying by e and integrating by parts gives

−βt −βt e Xt (φ) = e Xt (ψt ) Z t Z t −βs −βs = X0(ψ0) − βe Xs(ψs)ds + e dXs(ψs) 0 0 Z t Z −βs = X0(Pt φ) + e Pt−sφ(x)dM(s,x) (3.13) 0

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 −βt integral and hence we expect the limit of e Xt (φ) to be

Z ∞ Z −βs X0 (P∞φ) + e P∞φ(x)dM(s,x), (3.14) 0 which is a well deﬁned, ﬁnite random variable as

Z · Z Z ∞ −βs 2 −2βs e P∞φ(x)dM(s,x) < kφk e Xs(1)ds, 0 ∞ 0 which is ﬁnite in expectation. As P∞φ(x) does not depend on x, it follows that

Z ∞ Z −βs (3.14) = (P∞φ)X0(1) + (P∞φ) e dM(s,x) 0

= WP∞φ.

Given this decomposition for WP∞φ, we write ! 2 Z ∞ 2 −βt −βs P e Xt (φ) −WP∞φ ≤ 3P e P∞φ(x)dM(s,x) t

Z t Z 2 ! −βs 2 + 3P e (Pt−sφ(x) − P∞φ(x))dM(s,x) + X0(P∞φ − Pt φ) 0

If zt is a d-dimensional OU process satisfying dzt = −γzt dt + dBt , where Bt is

75 a d-dimensional Brownian motion, then

Z t −γt −(t−s)γ zt = e z0 + e dBt 0

−γt 1 −2γt and hence zt is Gaussian, with mean e z0 and covariance matrix 2γ (1 − e )I. 1 Evidently, z∞ is also Gaussian, mean 0 and variance 2γ I. We use a simple coupling: suppose that wt is a random variable independent of zt such that z∞ = zt + wt (i.e. −γt 1 −2γt wt is Gaussian with mean −e z0 and covariance 2γ e I). Then using the fact that φ ∈ Lip1 and the Cauchy-Schwartz inequality, followed by our coupling with z0 = x gives

Z 2 2 x X0(P∞φ − Pt φ) = E (φ(z∞) − φ(zt ))X0(dx) Z x 2 ≤ E (|z∞ − zt |) X0(dx)X0(1) Z x 2 = E |wt | X0(dx)X0(1) Z d = e−2γt |x|2 + X (dx)X (1) 2γ 0 0 Z −2γt 2 2 ≤ ce |x| X0(dx)X0(1) + X0(1) .

Taking expectations and using Cauchy-Schwartz and the assumptions on X0 gives exponential rate of convergence for the above term. R r R −βs Since we can think of 0 e Pt−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- R t R −βs nal element, 0 e Pt−sφ(x)dM(s,x). Note that this process is not in general a martingale in t. Therefore, we have

"Z t Z Z t Z 2# −βs −βs P e P∞φ(x)dM(s,x) − e Pt−sφ(x)dM(s,x) (3.15) 0 0

"Z t Z 2# −βs = P e (P∞φ(x) − Pt−sφ(x))dM(s,x) 0 Z t Z −2βs 2 ≤ P e (P∞φ(x) − Pt−sφ(x)) Xs(dx)ds . 0

76 Then as φ Lipschitz, by the coupling above,

Z t Z −2βs −2γ(t−s) 2 d (3.15) ≤ P e e |x| + Xs(dx)ds 0 2γ Z t Z −2βs−2γ(t−s) 2 d = e P |x| + Xs(dx) ds 0 2γ Z t −2βs−2γ(t−s) 2 d = e P Ks |Zs| + Xs(1) ds. 0 2γ

Applying the Cauchy-Schwartz inequality followed by Remark 3.3.10(b) gives 2 2 P Ks |Zs| = P |Zs| Ks(1);s < η 1 2 2 1 2 2 2 ≤ P |Zs| ;s < η P Xs(1)

1 1 2 2 ≤ B(s,γ,4) 2 P Xs(1) 1 2 1 βs 2 1 ≤ cB(s,γ,4) 2 e P X0(1) + X0(1) β where the last line follows by ﬁrst noting that

Z t Z −βt −βs e Xt (1) = X0(1) + e dM(s,x) 0 is a martingale. That is,

Z s 2! −2βs 2 2 −βr e P Xs(1) ≤ 2P X0(1) + e dM(r,x) (3.16) 0 Z · 2 −βr = 2P X0(1) + e dM(r,x) 0 s Z s 2 −2βr = 2P X0(1) + e Xr(1)dr 0 Z s 2 −βr −βr = 2P X0(1) + 2 e P e Xr(1) dr 0 Z s 2 −βr = 2P X0(1) + 2 e P(X0(1))dr 0 2 1 ≤ 2P X0(1) + X0(1) . β

77 Therefore,

" 1 # Z t 2 −2βs−2γ(t−s) βs 1 2 1 (3.15) ≤ e e B(s,γ,4) 2 P X0(1) + X0(1) ds 0 β Z t −2βs−2γ(t−s) βs d + e e P(X0(1))ds 0 2γ " 1 # Z t 2 −βs−2γ(t−s) 1 2 1 d ≤ e B(s,γ,4) 2 P X0(1) + X0(1) + P(X0(1)) ds 0 β 2γ

< Ce−ζ1t , where ζ1 = min(β,2γ) − ε where ε is arbitrary small and comes from the polynomial term in the integral. Finally,

Z ∞ 2! Z ∞ −βs 2 −2βs P e P∞φ(x)dM(s,x) = (P∞φ) P e Xs(1)ds t t Z ∞ 2 −βs −βs ≤ (P∞φ) e P e Xs(1) ds t 2 (P∞φ) −βt ≤ e P(X0(1)), β

−βs since e Xs(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 ∞ ζk/2 increasing sequence {tn}n=0 where {tn} ∩ [k,k + 1) = be c, for φ ∈ Lip1,

−βtn e Xtn (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 −βt e Xt (φ).

8 8 Lemma 3.4.7. Suppose γ > 0, P X0 |x| < ∞ and P X0(1) < ∞. If φ ∈ Lip1

78 and h > 0, then

4 h −β(t+h) −βt i 2 −ζ ∗t P e Xt+h(φ) − e Xt (φ) ≤ C(t)h e 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

−β(t+h) −βt e Xt+h(φ) − e Xt (φ) = X0(Pt+hφ − Pt φ) Z t+h Z Z t Z −βs −βs + e Pt+h−sφ(x)dM(s,x) − e Pt−sφ(x)dM(s,x) 0 0 Z t Z −βs = X0(Pt+hφ − Pt φ) + e ((Pt+h−s − Pt−s)φ(x))dM(s,x) 0 Z t+h Z −βs + e Pt+h−sφ(x)dM(s,x) t

≡ I1 + I2 + I3 where Pt is the OU semigroup. Using the Cauchy-Schwartz inequality, we can ﬁnd 4 bounds for P(|Ik| ),k = 1,2,3 separately.

4 4 |I1| = X0(Pt+hφ − Pt φ) 2 2 ≤ X0 (Pt+hφ − Pt φ) X0(1)

Recalling the simple coupling in the previous lemma to see that

79 I −2sγ −2tγ covariance matrix 2γ (e − e ). Therefore,

2 d (P φ(x) − P φ(x))2 ≤ |x|2 e−γ(t+h) − e−γt + e−2tγ − e−2(t+h)γ t+h t 2γ 2 d = e−2γt |x|2 1 − e−γh + 1 − e−2hγ . 2γ

Hence

4 P(|I1| ) " # 2 Z 2 −2γt −γh 2 d −2γt 2γh 2 = P e 1 − e |x| X0(dx)X0(1) + e 1 − e X0(1) 2γ

2 −4γt ≤ C1(d,γ)h e 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 R · R −βs 0 e ((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

4 2 P |I2| ≤ cP([N]t ) "Z t Z 2# −2βs 2 = cP e ((Pt+h−s − Pt−s)φ(x)) Xs(dx)ds 0 " !2 Z t 2 Z −2βs −γ(t+h−s) −γ(t−s) 2 ≤ cP e e − e |x| Xs(dx)ds 0 2 Z t −2βs ! # de −2(t−s)γ −2(t+h−s)γ + e − e Xs(1)ds 0 2γ " ! Z t 4 Z 2 −4βs −γ(t+h−s) −γ(t−s) 2 ≤ cP t e e − e |x| Xs(dx) ds 0 # 2 Z t 2 d −4βs −2(t−s)γ −2(t+h−s)γ 2 +t e e − e Xs(1) ds 2γ 0

80 " # Z t 4 Z 2 −2βs −γ(t+h−s) −γ(t−s) 2 −βs = ct e e − e P |x| e Xs(dx) ds 0

2 Z t 2 2 d −2βs −2(t−s)γ −2(t+h−s)γ −βs +t e e − e P e Xs(1) ds. 2γ 0

2 2 Since Xs(|x| ) = Ks(|Zs| ), by Remark 3.3.10 (b),

2 2 −2βs 2 2 2 −βs P e Xs(|x| ) ≤ P Zs e Xs(1) ;s < η

4 1/2 1/2 2 −4βs 4 ≤ P Zs ;s < η P e Xs(1) 1/2 1/2 8 −4βs 4 ≤ P Zs ;s < η P e Xs(1) 1/2 4 2 1/2 ≤ cB(γ,s,8) P X0(1) + sX0(1) + sX0(1) .

−4βs 4 The bound on the expectation of e Xs(1) follows by an application of the −βs R s −βr Burkholder-Davis-Gundy inequality to e Xs(1) = X0(1) + 0 e dM(r,x): ! 4 Z s 4 −βs 4 −βr P e Xs(1) ≤ cP X0(1) + e dM(r,x) 0

Z · 2! 4 −βr ≤ cP X0(1) + e dM(r,x) 0 s Z s 2! 4 −2βr ≤ cP X0(1) + e Xr(1)dr , 0

R t 2 where the last line follows as [M(φ)]t = 0 Xs φ ds. Then, using the Cauchy-

81 Schwartz Inequality followed by Fubini’s Theorem,

4 Z s −βs 4 −4βr 2 P e Xs(1) ≤ cP X0(1) + s e Xr(1) dr 0 Z s 4 −2βr −2βr 2 ≤ cP X0(1) + cs e P e Xr(1) dr 0 Z s 4 −2βr 2 1 = cP X0(1) + cs e P X0(1) + X0(1) dr 0 β 4 s 2 s ≤ cP X0(1) + X0(1) + X0(1) . 2β 2β 2 where in the third line we have substituted the calculation from (3.16). Therefore,

Z t 4 4 −2βs 1/2 −γ(t+h−s) −γ(t−s) P |I2| ≤ ct e B(γ,s,8) e − e 0 4 2 1/2 · P X0(1) + sX0(1) + sX0(1) ds Z t 2 ct −2βs −2(t−s)γ −2(t+h−s)γ 2 1 + 2 e e − e P X0(1) + X0(1) ds 4γ 0 β 1/2 4 2 1/2 ≤ ctB(γ,t,8) P X0(1) +tX0(1) +tX0(1) 4 Z t · e−γh − 1 e−2βse−4γ(t−s)ds 0 2 Z t ct −2γh 2 1 −2βs −4(t−s)γ + 2 e − 1 P X0(1) + X0(1) e e ds 4γ β 0

2 −ζ1t ≤ C2(t,γ,β)h e where C2 is polynomial in t. By another application of the BDG Inequality, and

82 noting that kφk = 1,

"Z t+h Z 4# 4 −βs P |I3| = P e Pt+h−sφ(x)dM(s,x) t " # Z t+h Z 2 2 −βs ≤ cP e Pt+h−sφ(x) Xs(dx)ds t

"Z t+h Z 2# −2βs ≤ cP e Xs(dx)ds t Z t+h −4βs 2 ≤ chP e Xs(1) ds t Z t+h −2βt h −2βs 2i = che P e Xs(1) ds t Z t+h −2βt 2 ≤ che P X0(1) + X0(1)/β ds t 2 −2βt ≤ C3(β)h e 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

4 −β(t+h) −βt 2 −ζ ∗t P e Xt+h(φ) − e Xt (φ) ≤ C(t,γ,β,d)h e

The following is a very useful result of Garsia, Rodemich, and Rumsey [7]. d Let Ψ : R → R, p : R+ → R be positive, continuous functions. Further, suppose Ψ is symmetric about 0 and convex with lim|x|→∞ Ψ(x) = ∞ and p(x) is increasing with p(0) = 0 .

Proposition 3.4.8. If f is a measurable function on [0,1] such that

ZZ f (t) − f (s) Ψ dsdt = B < ∞ (3.17) [0,1]2 p(|t − s|)

83 then there is a set K of measure 0 such that if s,t ∈ [0,1]\K then

Z |t−s| −1 B | f (t) − f (s)| ≤ 8 Ψ 2 dp(u). (3.18) 0 u

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 convergence of X˜t . Let d denote the Vasserstein metric on the space of ﬁnite measures d on R . I.e., for µ,ν ﬁnite measures, Z d(µ,ν) = sup φ(x)d(µ − ν)(x), φ∈Lip1

d recalling that Lip1 = ψ ∈ C(R ) : ∀x,y,|ψ(x) − ψ(y)| ≤ |x − y|,kψk ≤ 1 .

Theorem 3.4.9. Suppose X0 (1) < ∞ a.s. and γ > 0. Then on S,

a.s. d(X˜t ,P∞) −→ 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

−βtn can lay down an increasingly (exponentially) denser sequence e Xtn (φ) which converges almost surely, and that we can use Lemma 3.4.7 to get a modulus of

−βtn continuity on the process e Xtn (φ), which then implies that if the sequence is converging, then the entire process must be converging. 8 8 8 Assume that P K0 |y0| = P X0 |x| < ∞ and P X0(1) < ∞ and ar- −βt gue as in Theorem 3.4.1 in the general case. Let φ ∈ Lip1. Denote e Xt by 4 X˚t for the remainder of the proof. Let T > 0, and let Ψ(x) = |x| and p(t) = 1/2 3/4 λ 4 |t| log t where λ = e . Let BT (ω) be the constant B that appears in Proposition 3.4.8, with aforementioned functions Ψ and p, for the path X˚Tt (ω),t ∈ [0,1].

84 Then note that " ! # ZZ X˚Tt − X˚Ts P(BT ) ≡ P Ψ dsdt [0,1]2 p(|t − s|) h 4i ZZ P X˚Tt − X˚Ts = dsdt [0,1]2 3 2 λ |t − s| log |t−s| ZZ C(T(s ∧t))e−ζ ∗(s∧t)T 2|t − s|2 ≤ dsdt [0,1]2 3 2 λ |t − s| log |t−s| Z 1 Z t −ζ ∗(Ts) 2 2 C(Ts)e |t − s| = 2T dsdt 0 0 3 2 λ |t − s| log |t−s| Z 1 Z t 2 1 ≤ 2C(T)T dsdt 0 0 2 λ |t − s|log |t−s| C(T)T 2 ≤ 2e4 where C is the polynomial term that appears in Lemma 3.4.7. Since X˚t is continuous, by Garsia-Rodemech-Rumsey, for all s,t ≤ 1,

1 Z |t−s| B 4 ˚ ˚ T XTt − XTs ≤ 8 2 dp(u) 0 u 1 1 1 λ 2 ≤ AB 4 |t − s| 4 log T |t − s| where A is a constant independent of T (see Corollary 1.2 of Walsh [22] for this calculation). Rewriting the above,

1 1 λT 2 X˚t − X˚s ≤ DT |t − s| 4 log ∀s < t ≤ T, (3.19) |t − s|

1 4 BT 4 4 A TC(T) where DT ≡ A T . Note that P(DT ) = 2e4 , which is polynomial in T of ﬁxed degree d0 > 1. Let Ω0 be the set of probability 1 such that for all positive d integers T Equation (3.19) holds and DT ≤ T 0 for T large enough. To see P(Ω0) =

85 1, use Borel Cantelli: d0 4 4d0 P DT ≥ T = P DT ≥ T (D4 ) ≤ P T T 4d0 c ≤ T 3d0 which is summable over all positive integers T. −1 −1 2 T 8 λ d0 Suppose ω ∈ Ω0. Let δ (ω) be such that δ log δ = T . Then for all 1 integral T > T0(ω), and s,t ≤ T with |t − s| ≤ δ, X˚t − X˚s ≤ |t − s| 8 . ˚ Now let Xtn 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+ −ζk/2 ˚ (i.e. tn+1 −tn = ce for tn ∈ {tn} ∩ [k,k + 1)). Evidently Xtn converges a.s. to a limit X˚∞. Without loss of generality, assume convergence of the sequence on the set Ω0. −ζT/2 There exists T1(ω) such that for all T > T1, ce < δ. Hence, for all t such 0 0 −ζbtc/2 that T1 ∨ T0 < t ≤ T there exists tn ∈ {tn} such that |t −tn| < ce < δ and hence

˚ ˚ ˚ ˚ 0 ˚ 0 ˚ Xt − X∞ ≤ Xt − Xtn + Xtn − X∞ 1 0 8 ˚ 0 ˚ ≤ t −tn + Xtn − X∞ − ζbtc 16 ˚ 0 ˚ ≤ ce + Xtn − X∞ .

−βt Sending t → ∞ gives almost sure convergence of e Xt (φ) to X˚∞ = WP∞(φ) by 0 Theorem 3.4.5, since tn → ∞ with t. Note that this implies for φ ∈ Lip1

a.s. X˜t (φ) −→ P∞(φ) (3.20)

βt since on S, e → W −1 a.s. Kt (1) d d By Exercise 2.2 of [6] on M1(R ), the space of probability measures on R , the Prohorov metric of weak convergence is equivalent to the Vasserstein metric.

86 Note that the class of functions

n 2 −qi(x−xi) +bi Θ = ψ : ψ(x) = ∑ pi(x)e , i=1 d pi polynomial with rational coefﬁcients,qi ∈ Q+,bi ∈ Q,xi ∈ Q ,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,

X˜t (·) → 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.

0 Recall that Xt (·) = Kt (Yt ∈ ·) is the SOU process with attraction to its centre of mass.

0 Theorem 3.4.11. Suppose X0 (1) < ∞ a.s. and γ > 0. Then on S,

¯ a.s. ˜ 0 Y∞ d(Xt ,P∞ ) −→ 0 (3.21)

¯ Y∞ ¯ where P∞ represents the OU-semigroup at inﬁnity, with the origin shifted to Y∞.

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

˜ 0 ˜ Xt (φ) = Kt (φ (Yt )) Z t = K˜t φ Zt + γ Z¯sds 0 Z = φ (x + f (t,ω))dX˜t (dx)

R t ¯ ¯ ¯ a.s. where f (t) = γ 0 Zsds. Remark 3.3.4 (b) gives f (t) = Yt − Zt , and hence f (t) −→ Y¯∞ follows from Theorem 3.4.2. Note that

Z Z 0 Y¯∞ X˜ (φ) − P (φ) ≤ φ (x + f (t))dX˜t (dx) − φ (x + f (∞))dX˜t (dx) t ∞

Z Z + φ (x + f (∞))dX˜t (dx) − φ (x + f (∞))dX˜∞(dx)

≤ | f (t) − f (∞)| + d(X˜t ,X˜∞) since φ ∈ Lip1. Taking the supremum over φ and the previous theorem give

˜ 0 ˜ 0 ˜ ˜ a.s. d(Xt ,X∞) ≤ | f (t,ω) − f (∞,ω)| + d(Xt ,X∞) −→ 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 interactive 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: β ¯ (a) For 0 > γ > − 2 , Yt converges almost surely.

88 β ¯ (b) For 0 > γ > − 2 , Zt diverges exponentially fast. That is,

γt X lim e Z¯t → L 6= 0 S = 1. P 0 t→∞

2 Proof. Assume that P K˜ |y0| < ∞, 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, Y¯t will converge if P([Y¯ ]t ) < ∞ and which holds if the following quantity is bounded:

! ! Z ∞ |Y |2 −Y¯ 2 Z ∞ Y 2 s s ds S ≤ s ds S P s ; P s ; 0 eβ 0 eβ Z ∞ 1 + s6e−2γs ≤ c ds s 0 eβ < ∞, by Lemma 3.3.9 and by the conditions on γ. For (b), we require the following Lemma:

d Lemma 3.5.2. Let −β/2 < γ < 0 and X0 6= 0. For a measure m on R , let τa(m) d R be m translated by a ∈ R . That is, τa(m)(φ) = φ(x + a)m(dx). Then

(i) For all but at most countably many a,

γt ¯ Pτa(X0) e Zt → L 6= 0|S = 1. (3.22)

(ii) For all but at most one value of a

γt ¯ Pτa(X0) e Zt → L 6= 0|S > 0.

Proof of Lemma. We ﬁrst note that, by the correspondence (3.5), we have that

Z t −γt γs Z¯t = Y¯t − γe e Y¯sds. 0

89 Under our hypotheses Y¯t converges by Theorem 3.5.1 (a), and hence

Z t γt γs lim e Z¯t + γe Y¯sds = 0 (3.23) t→∞ 0 a.s. on S. Therefore, on S, γt lim e Z¯t exists a.s. (3.24) t→∞ Note that one can build a solution of (SE)2 with initial conditions given by (X ) Y 2 Y + a τa 0 by seeing that if t gives the solution of (SE)Y0,K, then t gives the 2 solution of (SE)Y0+a,K, and that the projection Z 0 Xt (·) = 1(Yt + a ∈ ·)Kt (dy) gives the appropriate interacting SOU process. By (3.23) and (3.24),

Z t γt ¯ c γs ¯ PX0 e Zt → L 6= 0 S = Pτa(X0) lim e Ysds = 0|S t→∞ 0 Z t K ( (Y )) γs s τa s = PX0 lim e ds = 0 S t→∞ 0 Ks(1) Z t K (Y ) γs s s = PX0 lim e a + ds = 0 S t→∞ 0 Ks(1) a Z t γs ¯ = PX0 − + lim e Ysds = 0 S . γ t→∞ 0

R ∞ γs ¯ The random variable 0 e Ysds 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.

d To complete the proof of Theorem 3.5.1 (b), choose a value a ∈ R 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,

γt τ (X ) X lim e Z¯t = 0,S = 0, P a 0 P 1 t→∞ holds, and hence by (3.25) we have

γt γt X lim e Z¯t = 0,S = X X lim e Z¯t = 0,S P 0 t→∞ P 0 P 1 t→∞ = 0.

γt Recalling from (3.24) that limt→∞ e Z¯t exists a.s., we are done.

β Note that for 0 > γ > − 2 , 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 Y¯t 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 exponentially fast, even while the COM of the interacting one settles down. This follows from the correspondence

Z t Y¯t = Z¯t + γ Z¯sds, 0 and the cancellation that occurs in the equation due to the exponential rate of Z¯t . The next lemma shows that Theorem 1 of Englander¨ 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):

β Lemma 3.5.3. On S, for the SOU process, X, with repulsion rate − d < γ < 0 and + d compactly supported initial measure µ, and any ψ ∈ Cc (R ) Z d|γ|t P e X˜t (ψ) −→ ξ ψ(x)dx, Rd 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 β ∞ d 0 < −γ < d . The theorem says that there is a function φc ∈ Cb R such that

X (ψ) P Wξ t −→ , µ (3.26) E (Xt (ψ)) µ(φc) where W is as in Lemma 3.3.7. + d Example 2 also shows that for ψ ∈ Cc R ,

−(β+γd)t µ lim e (Xt (ψ)) = µ(φc)m(ψ) t→∞ E

d where m is Lebesgue measure on R . Hence, manipulating the expression in (3.26) by using the previous equation and Lemma 3.3.7 gives

|γ|dt µ |γ|dt P ξW e E (Xt (ψ)) e X˜t (ψ) −→ lim µ(φc) t→∞ Xt (1) ξW e−(β+γd)t µ (X (ψ)) = lim E t −βt µ(φc) t→∞ e Xt (1) ξW µ(φ )m(ψ) = c . µ(φc) W

This lemma indicates that on the survival set, when γ < 0, one cannot naively normalize Xt by its mass since the probability measures X˜t are not tight. That is, a proportion of mass is escaping to inﬁnity and is not seen by compact sets. Note (β+γd)t that the lemma above implies that for Xt , the right normalizing factor is e .

Definition. We say a measure-valued process X˚ undergoes local extinction if for any finite initial measure X˚ and any bounded A ∈ B( d), there is a -a.s. finite 0 R PX˚0 stopping time τA so that X˚t (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 d undergoes local extinction (all the mass escapes). Hence for ψ ∈ Cc(R ), 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

Z t Z¯t = Z¯0 + Nt − γ Z¯sds, 0 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

Y¯t = Y¯0 + Nt , or in other words, the center of mass of the SOU process repelling from its COM is given by simply the fluctuations. d If one fixes a compact set A ⊂ R , then for the ordinary SOU process, X, A is exponentially distant from the COM of the process. However with X0, A will possibly lie in the vicinity of the COM of the process for all time. Therefore one 0 might expect that A is charged by a different amount of mass by Xt than Xt , and thus we might need to renormalize differently for the two cases. We finish with some conjectures:

0 Conjecture 3.5.5. On the survival set, if X0 is ﬁxed and compactly supported, then the following is conjectured to hold:

β (a) If 0 < −γ < d then there exists constant β + γd ≤ α < β so that for φ ∈ d Cc(R ), −αt 0 P e Xt (φ) −→ ν(ψ)

where ν is a random measure depending on Y¯∞.

0 (b) If β/d ≤ −γ then Xt 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 Z Z Z t 0 ¯ φ(x)dXt (x) = φ x + γ Zsds dXt (x) 0 still holds for t ﬁnite, the time integral of Z¯s 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 manner 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 superprocess 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 interactive 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 particles. 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 applications 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 equations (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- ˆ onal to Kt ) under the different measures PT . 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 repelling 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 attracting 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 desirable 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.

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