Spatial diffusions with singular drifts: The construction of Super Brownian motion with immigration at unoccupied sites

Transcription

1 Spatial diffusions with singular drifts: The construction of Super Brownian motion with immigration at unoccupied sites by Saifuddin Syed B.Math Hons., University of Waterloo, 24 AN ESSAY SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF Master of Science in THE FACULTY OF GRADUATE STUDIES Mathematics The University of British Columbia August 26 c Saifuddin Syed, 26

2 Abstract In this essay we explored two original problems related to diffusions with singular drifts. In Chapter, we construct diffusions X on [, using speed and scale analysis, that satisfy the stochastic differential equation SDE, X t X + X p s db s + bt, for < p 2, b, when X >, but exhibit sticky boundary behaviour at. I.e., X spends a positive amount time at zero, but never a full interval. We then find the SDE characterization of the sticky diffusions and show that in the case where p 2, the SDE fails to uniquely encode the boundary behaviour of the process. In Chapter 2, we then proceed to the realm of spatial diffusions and construct a nontrivial continuous measure-valued process X that solves the stochastic partial differential equation SPDE, X t X 2 + XẆ + A, where W is a space-time white noise and A is a continuous measure-valued immigration such that A t is only active in the regions where X t does not occupy mass. ii

3 Acknowledgements First and foremost, I am eternally grateful to my supervisor Dr. Edwin Perkins for sharing his tremendous insight and intuition with me throughout my Masters. Working with him has given me the opportunity to learn an aspect of mathematics that I have always found fascinating and beautiful, but felt was out of my reach. Without his foresight and hard work, this essay would not have been possible. I also would like to thank Fok Shuen Leung for giving me the opportunity to cultivate my love of teaching mathematics. I am honoured to have had the opportunity to share my passion with such a remarkable group of students over the last two years. I am sure they will go on to do great things despite me scarring them. Finally, I would like to express my sincerest gratitude to my family and friends I have had the honour of conversing with throughout my Masters. Their guidance and support has allowed me to grow both academically and personally and I am forever in their debt. They are the reason the two years have been an unforgettable part of my life. Saifuddin Syed The University of British Columbia August 26 iii

4 Preface Over the course of my masters, I have worked on a variety of problems in stochastic analysis and diffusions. In this essay we will work through two of them; each self-contained in their own chapter. In Chapter, we study a class of dimensional diffusions on X on [, which satisfy the stochastic differential equation SDE, X t X + X p s db s + bt, when X >, for p 2, b, and exhibit sticky boundary behaviour at. By sticky boundary behaviour, we mean that the process spends a positive time at but never a full interval. We will construct the diffusions using scale and speed analysis and determine when their SDE representation uniquely encodes the sticky boundary behaviour. In Chapter 2, we turn our attention to spatial diffusions. Motivated by population genetics, we construct a population on R, undergoing random motion and critical reproduction, under the influence of immigration at unoccupied sites. In particular, we construct a measure-valued diffusion X satisfying the stochastic partial differential equation SPDE, X t X 2 + XẆ + A. Where W is a space-time white noise and A is a continuous measure-valued process which is only supported at the location where X occupies no population. This will be formalized later. iv

5 Contents Abstract Acknowledgements Preface Contents ii iii iv v SDE characterization of diffusions with sticky boundaries. Characterization of sticky behaviour for < p < Existence of sticky boundary behaviour Differentiability of random time change Equivalence of SDE to speed/scale Non-uniqueness of SDE for sticky Bess 2 α process Summary Super Brownian motion with immigration at unoccupied sites 9 2. Notation Construction and SPDE characterization of X ε Tightness Proof of Theorem 2.3.4: C-relative compactness of {X εn φ}, and {A εn φ} Proof of theorem 2.3.4: Compact containment condition v

6 CONTENTS CONTENTS 2.4 SPDE characterization of limit points Density of X Proof of Theorem 2.5.: C-relative compactness of {u ε } Domination of Immigration Non-triviality of X Contribution of long living populations to X ε Proof of Non-triviality of X Summary Bibliography 75 vi

7 Chapter SDE characterization of diffusions with sticky boundaries In Theorem 8 of Burdzy et al., 2, Burdzy, Mueller, and Perkins showed that when p < 2, b >, non-negative solutions to the stochastic differential equation SDE, X t X + exhibit Feller s sticky boundary behaviour at. I.e, X p udb u + bt,. X s ds >. Our goal in this chapter is to strengthen their result and classify all diffusion on X on [, that satisfy. when when X > and exhibit Feller s sticky boundary behaviour at. We will do this by following the method of Ito and McKean outlined in Knight, 98, 6,7. We will first examine the case where p < 2. We note that we separating the drift bt into time spent by the process away from, and the spend at to rewrite. as, X t X + X p udb u + b X s > ds + b X s ds..2 Section., will show that the stickiness of X is controlled by the coefficient of the X term in the drift in.2. We will use speed and scale analysis to argue that the SDE fully characterize the sticky boundary of our process.

8 In the case of p 2, after scaling by a constant. is equivalent to the equation, X t X + 2 X u db u + αt,.3 which is the SDE of a Bess 2 α process. We will construct a family of sticky Bess 2 α processes and find their corresponding SDE. We will then argue that in this case the SDE representation is not robust enough to uniquely encode the boundary behaviour.. Characterization of sticky behaviour for < p < 2 Let X t be a non-negative solution to the SDE, X t X + X p s db s + b X s > ds + ρ X s ds,.4 where < p < 2, and b, ρ >. Note that in the special case where b ρ, we get.. Our aim for this section is to prove the following: X exhibits sticky boundary behaviour at governed by ρ. 2 The SDE encodes the boundary behaviour as described by the scale function and speed measure. We will show by analysing the scale function, and speed measure for X, and 2 by show that any diffusion on [, with the same scale function and speed measure as X, must satisfy.4. We now proceed with the first step of the program... Existence of sticky boundary behaviour Proposition... If X is a solution to.4, then it is the diffusion on [, with the scale function, sx x { } 2b y 2p exp dy,.5 2p 2

9 and, speed measure m on [, s defined by, mdx Moreover if T inf{t X t }, then for all ε >, dx s s x 2 s x 2p + ρ δ dx..6 T +ε T X s ds >..7 Remark..2. Note that the scale function can be found by solving the ode in terms of the drift and volatility as described in Rogers and Williams, 2, V.28. Also since b > and < p < 2, we have s is increasing with s < and has an inverse s : [, s [,. 2. The proof below is a modification of the proof of Theorem 8 in Burdzy et al., Proof. We begin by analysing the scale function, So when x > we have the s is given by, { } 2b x s 2p x exp. 2p { } 2b x s 2p x exp 2bx 2p 2bs xx 2p. 2p Define L X t x as the local time process of X, and let Y t sx t. So we have by Ito- Tanaka s formula, Y t sx t Y + + ρ s X s Xs p db s + b s X s X s > ds s X s X s ds + L X t xds x. 2 3

10 Now let s simplify the local time term. Since s is continuous, we have ds has no atoms. Thus we have Let us define 2 L X t xds x Y t Y + b b x > L X t xds x x > L X t xs xdx x > L X t x 2bs xx 2p dx s X s X p s db s + ρ U Y X s > s X s Xs 2p Xs 2p ds X s > s X s ds. s X s X s ds. s X s 2 X 2p s ds. Define the random time change α : [, U [, as the solution to the equation αt s X s 2 X 2p s ds t..8 It is clear that since the integrand is non-negative we have that α is increasing and positive. Since X is non zero on any interval, α is continuous. Define R t Y αt, we will now show that R is a reflecting Brownian motion starting at Y for t < U, and can be extended to in the natural way for t U. We have for t < U, R t Y + αt Y + β t + A t. s X s X p s db s + ρ αt X s ds Where β t Y + αt X p s db s, and A t ρ αt X s ds. Note that for t < U we have by.8, β t αt s X s 2 X 2p s ds t. 4

11 So β is a stopped Brownian motion starting at Y, and A is a continuous non-decreasing and supported by {t : X αt } or equivalently {t < U : R t }. Thus by the Skorokod problem Rogers and Williams, 2, V.6, R is a reflecting Brownian motion and for t < U, ρ αt X s ds A t L R t..9 Note that.9 implies.7. Thus it remains to find the speed measure of X. Since by the fundamental theorem of calculus, We now note that by.9 and., t If we let s αu, we get t α t α t s Therefore the speed measure of X is s X s 2 X 2p s ds α t, α t s X t 2 X 2p t.. X s > ds + X s ds X s > s X s 2 Xs 2p dα s + ρ LR α t. X αu > s X αu 2 X 2p αu du + ρ LR α t Ru > s s R u 2 s R u 2p du + ρ LR α t [ L R α t x x > dx s s x 2 s x 2p + ] ρ δ dx. mdx x > dx s s x 2 s x 2p + ρ δ dx, on [, s as required. So we can conclude that X t s R α t, 5

12 is the diffusion on [, with scale function s and speed measure m, where t s L R α t xmdx. Therefore we have shown that solutions to the SDE.4 exhibit sticky boundary behaviour at. The goal now is to show that the converse to Proposition.. holds as well...2 Differentiability of random time change In order to show the converse of Proposition.., we will be forced to work with various random time changes by τt, which is determined as the solution to the equation, t τt fb s ds + ρ L t, where B is a standard Brownian motion with local time L, and sufficiently f positive and nice. derivative. The goal of this section is to show that τ is a.s. differentiable and to compute it s Lemma..3. Let L be the local time of Brownian motion, then, Proof. Let s show that PL 2 n lim inf h L h h, < φ2 n i.o., for φh h /2 log/h ε, where ε >. So indeed, using the fact that L t has the same law as sup s t B s, and the reflection principle we have, 6

13 PL 2 n < φ2 n P B 2 n < φ2 n P 2 n B < 2 n log2 n ε log 2 ε P B < n +ε log 2 ε n 2 +ε e z2 2 dz 2π log 2 ε 2 n +ε. Which is summable. Thus by Borel-Cantelli, PL 2 n < φ2 n i.o. and lim inf h L h lim inf φh n L 2 n φ2 n. Where the first equality is using the fact that L and φ are continuous, along with an interpolation argument. Now we note that lim inf h L h h lim inf h L h φh φh h lim φh h h. Proposition..4. Let f be a non-negative integrable function such that for all η >, f η Define τt to be the continuous inverse of, At inf fx >. η< x <η fb s ds + ρ L t, Where L t is the local time of Brownian motion. Then the following holds: a τt is a.s. absolutely continuous with respect to the Lebesgue measure. b If f is continuous when x >, then the Radon-Nikodym derivative equals τ t f B τt Bτ t >. 7

14 Proof. a Since f is non-negative, and τ is the inverse of A, we have τ is increasing and satisfies, t τt fb s ds + ρ L τ t.. We will show that τt is absolutely continuous with respect to Lebesgue measure on [, T ] for all T >. Let ε > and suppose [s i, t i ] [, T ] satisfying t i s i < δ. Now note that, τt i τs i i i i + i τt τti τs i τti du τs i τti τt + τs i I + I 2 + I 3 B u ηdu + i B u η du B u ηdu + i B u η du i τti τs i τti τs i η < B u < η du η < B u < η du Since B t is continuous on [, T ], we have that there exists ηω small enough such that B t ω ηω. Therefore I 3. Because Brownian motion occupies zero time at zero, we can pick η small enough such that I < ε 2. So it remains to bound I 2. Let δ < fηε 2, then we have 8

15 I 2 i i f η τti τs i τti i τs i τti τs i η < B u < η du fb u η < B u < η du f η fb u du t i L f τti s i L τsi, by. η i t i s i f η f η δ i < ε 2 Thus we have shown τ is absolutely continuous with respect to Lebesgue measure. By Radon-Nikodym theorem, there exists a τ t such that τt b By rearranging., we see that for h >, τ udu. τt+h τ t fb s ds + h ρ L τt+h L τt. h Suppose that B τt, so the second term is non-zero. Since f we have the inequality By rearranging terms we have, ρ L τt+h L τt. h ρ L τ t+h L τt h L τ t+h L τt τ t+h τ t L τ h τ h τ t+h τ t h τ t+h τ t θ τt..2 h 9

16 Where θ t is the left shift by t. The last equality was applying the shift to the Markov process Lτ t τ t by the stopping time τ t. The strong Markov property tells us the first term of.2 has the same law as Lτ h τ h. So by lemma..3, the liminf is infinity, which implies the limit is also infinity as h goes to zero. So for all N there is a δ > such that h < δ implies, Therefore we have, L τh τ h θ τt > N. ρ N τ t+h τ t. h Letting N go to infinity gives us that the right derivative of τt both exists and equals. Since τ is absolutely continuous with respect to Lebesgue measure, we have the left limit also exists and equals the right. So when B τt, we have τ t. Now Suppose that B τt >, then L τt+h L τt for h small enough. So we have τt+h τ t fb s ds h Therefore by rearranging the terms we get, τ t+h τ t lim h h τt+h τ t fb s ds τ t+h τ t τ t+h τ t. h τ t+h τ lim τt+h h τ t fb s ds fb τt. The last equality was by the fundamental theorem of calculus. Thus we have shown that τ t B τt B τt >...3 Equivalence of SDE to speed/scale Equipped with Proposition..4, we are ready to prove the converse to Proposition... Proposition..5. Suppose X is a diffusion on [, with s, and speed measure m defined on [, s given by.5, and.6 respectively. Then X is a solution to.4.

17 Proof. Let R be a reflected Brownian motion with initial law sx with local time L R. We have by the definition of scale function and speed measure that X t s R τt, where τ t is the continuous inverse of At s s L R t xmdx L R t xx > s s x 2 s x 2p dx + ρ LR t R u > s s R u 2 s R u 2p du + ρ LR t..3 Since sx x + ox, and s x + ox, we have s x x + ox and x > x > s s x 2 s x 2p x 2p, Which satisfies the criteria of Proposition..4, thus τ t s X t 2 X 2p t X t >..4 Now to get an SDE representation of X we apply Ito-Tanaka s formula. Xt s R τt s R + τt s sx + X + τt s R u dr u + L R τ 2 t xds x s s R u dβ u + L R u + τt s R u du 2 τt s s R u dlr u + τt 2 τt s s R u dβ u + s R u du X + M t + I t + I 2 t..5 Where β t is a Brownian motion. We will have to take care of each of the three terms

18 individually. Let s begin with analysing the quadratic variation of M: [M] t τt s s R u 2 du s s R τv 2 τ vdv s X v 2 s X v 2 X 2p v X v > dv X 2p v X v > dv X 2p v dv. The third line was because of.4. Define B X u > Xu p dm u + where β is an independent Brownian motion and note B t X u > Xu 2p Xu 2p du + X u d β u, X u du t. So by Lévy s characterization of Brownian motion, there exists a Brownian motion B, such that M t We can finally deal with the finite variation terms. I t τt τt τt L R τ t. X p v d B v..6 s s R u dlr u s s dlr u dl R u 2

19 We now note that by rearranging.3 we get, τt It ρ t ρ ρ t t ρ t ρ R u > s s R u 2 s du R u 2p R τv > s s R τv 2 s R τv 2p τ vdv X v > s X v 2 Xv 2p s X v 2 Xv 2p dv X v > dv X v dv..7 Before we simplify I 2, let s first compute s x, and we use the fact that when x >, we have s x 2bs xx 2p. So for x >, s x s s x As s is continuous at x, we have I 2 t 2 2 b b τt [s s x] s s x 2 s s x s s x 3 2bs s xs x 2p s s x 3 2b s s x 2 s x 2p. 2b s s R u 2 s du R u 2p 2b s s R τv 2 s R τv 2p τ vdv s X v 2 Xv 2p X v > s X v 2 X 2p v dv X v > dv..8 By substituting.6,.7,.8 into.5, we have shown that X t X + X p v d B v + ρ X v dv + b 3 X v > dv,

20 as required. This implies that for the case where p < 2,.4 encodes the same information as the scale function and speed measure..2 Non-uniqueness of SDE for sticky Bess 2 α process In the case where p 2, we have. is equivalent to the SDE for the Bess2 α process given by, X t X + 2 X u db u + αt, for α >. The analysis of the Bess 2 α process in Rogers and Williams, 2, V.48 shows that is recurrent if and only if < α < 2, but it does not exhibit any sticky boundary behaviour, i.e., for all t >, X u du. Following the method of Ito and KcKean as described in Knight, 98, 6,7, we can construct a Bess 2 α process with sticky boundary behaviour at, by applying the same construction as in the case of p < 2. It was shown in Rogers and Williams, 2, V.48 that the Bess 2 α process has scale function sx x 2 α 2, and speed measure m defined on [, given by mdx 2 α 2 x 2α 2 2 α x > dx. So if X is the Bess 2 α process with sticky boundary behaviour at, then the X also has the scale function s and speed measure m given by mdx mdx + ρ δ dx, where ρ > controls how sticky is. We will now find an SDE representation of X, the sticky Bess 2 α process. 4

21 Proposition.2.. The process X on [, with the scale function and speed measure on [,, given by sx x 2 α 2, and, mdx respectively, satisfies the SDE: X t X + 2 α 2 x 2α 2 2 α x > dx + ρ δ dx, 2 X u db u + α X u > du..9 Proof. Let B be a Brownian motion with initial law sx X 2 α 2, and L t x be the local time of the reflected Brownian motion B. Let s define At L t xmdx L t x 2 α 2 x 2α 2 2 α x > dx + ρ δ dx 2 α 2 B u 2α 2 2 α Bu > du + ρ L t. Let τ t denote the right continuous inverse of A, ie, τ t satisfies, t τt 2 α 2 B u 2α 2 2 α Bu > du + ρ L τ t. Let Y t B τt be the scaled sticky Bess 2 α process. So we have is the unscaled Bess 2 α. 2 X s Y t B τt 2 α.2 Since x p is convex and has an integrable second derivative when p >, and 2 2 α >, we can apply Tanaka s formula for x 2 2 α to.2. X t B τt 2 2 α X 2 α τt α α sgnb u B u α 2 α dbu + τt 2 2 α 2 α 2 α B u 2α 2 2 α du X + M t + I t..2 5

22 Now let us simplify each term individually, but first we will need τ t. By the Proposition..4, τ t 2 α 2 B τt 2 2α 2 α Bτt > 2 2α 2 α 2 α 2 Yt Y t > 2 α 2 X 2 α 2 t 2 2α 2 α Xt > 2 α 2 X α t X t >..22 To determine the quadratic variation of M, we will make the substitution u τv and using.22 Define M t τt β t 4 2 α 2 B u 2α 2 α du 4 2 α 2 B τ v 2α 2 α τ vdv 4 2 α 2 α 2 X 2 v 2α 4X v X v > dv. X v > 2 X v dm v + 2 α 2 α 2 X α X v > dv v X v dw v, where W is an independent Brownian motion. β has quadratic variation, β t Therefore β is a Brownian motion and M t X v > 4X v 4X v dv + X v > 2 X v dβ v X v dv t. 2 X v dβ v..23 6

23 Finally we simplify I t, using the substitution u τ v and using.22. I t 2 α τt 2α 2 α 2 B u 2α 2 2 α du α 2 α 2 B τ v 2α 2 2 α τ vdv α 2 α 2 α 2 X 2 v 2α 2 2 α 2 α 2 X α X v > dv v X v > dv..24 Finally by substituting.23, and.24 into.2 we conclude, that X satisfies the SDE, X t X + 2 X v dβ v + α X v > dv. For each ρ >, let X ρ be the diffusion on [, with scale function sx x 2 α 2, and speed measure, m ρ dx 2 α 2 x 2α 2 2 α x > dx + ρ δ dx. Proposition.2. tells us that each X ρ satisfies.9. following. Therefore we have proven the Corollary.2.2. The solutions to the SDE.9 are not unique in law. This analysis for the Bess 2 α process shows that SDE s are not in general best tool to analyse the boundary behaviour of diffusion..3 Summary In Chapter, we first began by analysing the the SDE, X t X + X p s db s + b X s > ds + ρ X s ds..25 7

24 for < p < 2, and b, ρ >. This was motivated Burdzy et al., 2 where they studied the same SDE when ρ b, where they discovered interesting sticky boundary behaviour. In Section., we showed that non-negative solution X to.26 exhibits sticky boundary behaviour governed by ρ. We did this in Proposition.. by analysing the scale function and speed measure of X. We went a step further and showed in Proposition..5 that any diffusion on [, with the same scale function and speed measure as X must also be a solution to.26. Afterwards we moved our attention to the case where p 2, to the Bess2 α for < α < 2. In Section.2, we followed the construction of Ito and McKean to construct a family of diffusions X ρ on [, that behaved like Bess 2 α with sticky boundary behaviour at governed by ρ. In Proposition.2., we found that each X ρ was a solution to the SDE, X t X + 2 X s db s + α X s > ds..26 This showed that the SDE s in general are not robust enough to encode the boundary behaviour of diffusions, and demonstrates the utility of speed and scale analysis. 8

25 Chapter 2 Super Brownian motion with immigration at unoccupied sites Let us examine.4 in the case where b. This results in the SDE, X t X p s ds + ρ X s dx. 2. Note that X is a non-trivial diffusion on [, with a continuous immigration only active when the process is zero. Using 2. as a motivation, we will explore this idea for spatial processes. A one-dimensional super-brownian motion SBM, X is used in population genetics to model a population on a line undergoing random motion and critical reproduction. Formally, X t is a random finite measure on R changing continuously in t, and represents how the total population is dispersed at time t. Perkins, 22, III.,4 showed that at each time t, the population X t has compact support, and X has satisfies the stochastic partial differential equation SPDE, X t X 2 + XẆ. where W is space-time white noise. Perkins, 22, III. also showed that for all t, the population X t is a.s. compactly supported. We are interested in seeing what happens to the population represented by SBM when we allow a continuous immigration only at the 9

26 locations where the population does not occupy space. Our goal in this chapter will be to construct such a process and classify it. We will define the spatial process X ε as the sum of independent SBM populations X i of initial mass ε generated at some random point x i generated by a distribution ψxdx at time t i via a Poisson process with temporal rate ε dt. We will only allow Xi to contribute to X ε when the previously contributing populations do not occupy space at x i. Note as ε gets smaller, each SBM cluster X i will potentially add ε mass to X ε but there will be proportionally ε clusters contributing. This suggests that as ε, our process should converge to some non-trivial X. The plan of attack will be as follows; after making our construction of X ε more formal, we will then show that {X ε } are tight as ε. This will give us the existence of a continuous measure-valued process X. We will then show that X is non-trivial and satisfies a SPDE of the form, X t X 2 + XẆ + A. Where A is a continuous measure-valued immigration that is only active when the process X occupies no space. Without further ado, let us begin our construction. 2. Notation Let Ξ be a F t -Poisson point process PPP on R + R d with rate Λdt, dx λdtψxdx, where λ >, and ψ is a bounded density on R d. Given A R + R d, we denote ΞA PoiΛA to be the number of points generated by Ξ in A. Furthermore, define ˆΞ Ξ Λ, to be the compensated F t -PPP for Ξ. It will be useful to refer to all the points generated by Ξ up to time t. We will denote this by the following process, Ξ t Ξ[, t] R d. 2

27 We similarly define Λ t and ˆΞ t. By Ikeda and Watanabe, 24, III.3, ˆΞt is a F t - martingale in t with the previsible square function ˆΞ t Λ t and quadratic variation [ˆΞ] t Ξ t, i.e. the number of jumps of ˆΞ until time t. Suppose f : R + R d Ω R is square integrable and F t -predicable and f t x is Borel-measurable. We denote the integral of f with respect to Ξ by, A t f f s xξds, dx t i tf ti x i. t i,x i Ξ If A t f < a.s., then the compensated integral, Â t f f s xˆξds, dx f s xξds, dx f s xλds, dx, is the stochastic integral of f with respect to ˆΞ. By Ikeda and Watanabe, 24, III.3, we know that Ât is a F t -martingale with the previsible square function, Âf t f s x 2 Λds, dx, and quadratic variation, [Âf] t A t f f s x 2 Ξds, dx. Finally, it will be useful to define common spaces that will be used throughout this paper. We define, C b {f : R R f is bounded and smooth}, C c {f : R R f is smooth with compact support}, { } C f : R R f is continuous, lim fx, x M F {µ µ is a finite measure on R}. Given a metric space S, d, define CS and DS as, CS {f : R + S f is continuous}, DS {f : R + S f is cádlág}. 2

28 We equip CS with the topology of uniform convergence on compact sets, and DS with the Skorokhod J topology. In this paper we be interested in the case where S is either R, C, or M F. 2.2 Construction and SPDE characterization of X ε For ε >, let Ξ ε be a PPP on R + R with rate Λ ε dt, dx ε dtψxdx, where ψ is a bounded, compactly supported density on R. Further suppose that φ : R R + is a Lipschitz function supported on [, ] satisfying φ, and φ. We define φ x ε by φ x εy y x εφ. ε Remark Note that the compactness of ψ is not necessary and is only used to simplify some of the analysis in the proof of Theorem The theorem is still true for ψ bounded but requires more effort. Let t i, x i be the points generated by Ξ ε, and {X i } be independent SBM s with initial law φ x i ε zdz in chronological order. By Theorem III.4.2 in Perkins, 22, III.4 have for each X i, there exists process u i with sample paths in CC such that for all Xtdx i u i txdx. Suppose κ. We define κ i inductively by, κ i κ j u j t i t j x i. 2.2 t j <t i So κ i if and only if, for all j < i, κ j implies SBM X j t i x i. We define M F -valued processes X ε and X ε by, does not occupy space at X ε t t i t κ i X i t t i, 2.3 and, X ε t si t X i t t i,

29 respectively. Note that X ε and X ε are processes with sample paths in DM F. Since Xtdx i u i txdx, we immediately have Xt ε dx u ε txdx for the C -valued process u ε defined by u ε t κ i u i t t i t i t Note that u ε has sample paths in DC and if then 2.2, and 2.5 give us, u t x lim s t u sx, κ i u ti x i. 2.6 It will also be useful to define the natural filtration F ε t and F ε t by, respectively. F ε t σx ε s, X ε s, Ξ ε s s t. Again, by Theorem III.4 in Perkins, 22, III.4 there are independent F ε t -white noises W i on R +, R, such that for all φ Cb, Xi satisfies the SPDE, φ t Xtφ i Xs i ds + φx u 2 i sxw i ds, dx + φ, φ x i ε L i tφ + M i t φ + A i tφ. 2.7 Where M i φ is a continuous F ε t -martingale with M i φ t Xi sφ 2 ds. By substituting 2.7 into 2.3 and 2.4, we get the following SPDE characterization of X ε and X ε. Theorem For all φ C b, a There is a continuous F ε t -martingale M ε φ with M ε φ t Xε sφ 2 ds, such that X ε satisfies the SPDE, φ Xt ε φ Xs ε ds + Mt ε φ + 2 φ, φ x ε u ε s x Ξ ε ds, dx 2.8 L ε tφ + M ε t φ + A ε tφ

30 b There is a continuous F ε t -martingale M ε φ with M ε φ t X ε sφ 2 ds, such that X ε satisfies the SPDE, X ε t φ X ε s φ 2 ds + M ε t φ + φ, φ x ε Ξ ε ds, dx 2. L ε tφ + M ε t φ + Āε tφ. 2. Proof. a Suppose t i, x i are the points generated by Ξ ε. We begin by substituting 2.7 into 2.3. X ε t φ t i t κ i X i t t i ti κ i t i t t i t ti X i s κ i X i s φ 2 φ 2 ds + Mt t i i φ + φ, φ x i ε ds + κ i Mt t i i φ + κ i φ, φ x i ε L ε tφ + M ε t φ + A ε tφ. 2.2 It remains to show that L ε φ, M ε φ, and A ε φ, satisfy 2.8. We will deal each sum individually. To simplify L ε φ we exchange the sum and integral. L ε tφ t i t ti t i s Xs ε κ i X i s φ 2 φ κ i Xs t i i 2 φ ds φ t i sκ i Xs t i i t i t 2 φ t i sκ i Xs t i i 2 t i t ds 2 ds ds ds

31 Also note that by 2.6, A ε tφ t i t κ i φ, φ x ε t i t u ε t i x i φ, φ x i ε u ε s x φ, φ x ε Ξ ε ds, dx. 2.4 It remains to show that M ε t φ t i t κ im i t t i φ satisfies the properties of the theorem: i M ε φ is continuous, ii M ε φ is a martingale, iii M ε φ t Xε sφ 2 ds. We will deal with each one at a time. i Note that M i φ are continuous F ε t -martingales starting from, and since M ε φ is a sum of M i φ, we have M ε φ must also be continuous. ii Since M i φ is a F ε t -martingale, for t t i, M i t t i φ is F ε t t i -measurable, and hence F ε t -measurable since F ε t t i F ε t. Also since Ξ ε t is F ε t -measurable, we can conclude M ε t φ t i t κ im i t t i φ is F ε t -measurable. To show the martingale property, note that we can decompose M ε t φ as, M ε t φ t i t κ i M i t t i φ ti κ i t i t κ i t i t t i φx u i sxw i ds, dx φx u i v t i xw i dv, dx

32 For s t, we separating 2.5 further into the time before s and after s. Mt ε φ s κ i φx u i v t i xw i dv, dx t i s t i + κ i φx u i v t i xw i dv, dx t i s s + κ i φx u i v t i xw i dv, dx s<t i t M ε s φ + t i s t i κ i s φx u i v ti xw i dv, dx + s<t i t κ i M i t t i φ. 2.6 We now take conditional expectation of the second and third terms in 2.6 with respect to F ε s. Note that when t i s, κ i is F ε s -measurable and for all n N, Ξ ε s n is Fs ε -measurable. This coupled with the fact that s φx u i v t i xw i dv, dx is a Ft ε -local martingale for s t, gives us E κ i φx u i v t i xw i dv, dx F ε s ti s s E Ξ ε s nκ i φx u i v t i xw i dv, dx F ε s i n s Ξ ε s nκ i E φx u i v t i xw i dv, dx i n s. 2.7 To deal with s<t i t κ im i t t i φ, note that Ξ ε t Ξ ε s is independent of F ε s, and since s < t i t, we have M i t t i φ t i φx u i v t i xw i dv, dx and κ i are independent of Fs ε. So the third term of 2.6 is independent of Fs ε, and thus, E κ i Mt t i i φ E κ i Mt t i i φ. 2.8 s<t i t F ε s s<t i t Note that M i φ is independent of κ i, so by conditioning on κ i we get F ε s E κ i M i t t i φ E κ i EM i t t i φ

33 Since s<t i t κ imt t i i φ is the sum of Ξ ε t Ξ ε s independent random variables with mean, Wald s equation gives us, E κ i Mt t i i φ E Ξ ε t Ξ ε s E κ i Mt t i i φ t s. 2.2 ε s<t i t Combining 2.6, 2.7, 2.8, and 2.2, we get, EM ε t φ F ε s M ε s φ. Thus M ε φ is a F ε t -martingale. iii We define N ε t φ M ε t φ 2 Xε uφ 2 du. In order to show M ε φ t Xε uφ 2 du, we need N ε φ to be a F ε t -martingale. Since M ε t φ is a F ε t - measurable, so is N ε t φ. Before we prove the martingale property, it will be useful to define N i φ to be the square F ε t -martingale for M i φ. By definition of M i φ, Note that for t t i, N i t φ M i t φ 2 M i φ t M i t φ 2 M i φ t ti ti X i uφ 2 du We simplify Xε uφ 2 du by using 2.2, X ε uφdu t i u X i uφ 2 du. t i ux i u t i φ 2 du. 2.2 κ i Xu t i i φ 2 du κ i t i uxu t i i φ 2 du t i t κ i t i uxu t i i φ 2 du t i t t i t κ i M i φ t ti

34 We can now use 2.22 to simplify N ε φ t. Nt ε φ Mt ε φ 2 X ε uφ 2 du κ i Mt t i i φ κ i M i φ t ti ti t t i t κ i M i t ti φ 2 M i φ t ti + 2 t i t κ i Nt t i i φ + 2 t i t 2 t j <t i t κ j κ i M j t t j φmt t i i φ t j <t i t κ j κ i M j t t j φm i t t i φ For s < t, we further decompose 2.23 into the clusters born before and after time s. Nt ε φ κ i Nt t i i φ + 2 κ j κ i M j t t j φmt t i i φ t i s t j <t i s + κ i Nt t i i φ + 2 κ j κ i M j t t j φmt t i i φ t j <t i,s<t i s<t i t Similar to the proof of ii, when t i s, κ i is Fs ε -measurable and for all n N, Ξ ε s n is Fs ε -measurable. Also, Nt t i i and M j t t j φmt t i i φ are a Ft ε - martingales for t t i, since M j φ is independent of M i φ. So the conditional expectation of the first term in 2.24 becomes, E κ i Nt t i i φ κ i Ξ ε s n ENt t i i φ Fs ε ti s i n F ε s n κ i Ξ ε s n N i s t i φ n i n t i s κ i N i s t i φ Similarly the conditional expectation of the second term becomes, E 2 κ i κ j M j t t j φmt t i i φ 2 κ i κ j M j s t j φms t i i φ. Fs ε t j <t i s t j <t i s

35 For the other two terms of 2.24, again, as in ii, note that Ξ ε t Ξ ε s is independent of F ε s, and since s < t i t, N ε t t i and M i t t j are independent of F ε s, we have E s<t i t E s<t i t κ i Nt t i i φ + 2 κ i κ j M j t t j φm i t t i φ F ε s t j <t i,s<t i κ i Nt t i i φ + 2 κ i κ j M j t t j φmt t i i φ 2.27 t j <t i,s<t i In the last line we used the fact that each term in the random sums in 2.27 has mean along with Wald s equation, as previously done in 2.9. Therefore combining 2.24, 2.25, 2.26, and 2.28, we have shown EN ε t φ F ε s t i s κ i N i s t i φ + 2 t i <t j s κ i κ j M i s t i φm j s t j φ N ε s φ. Thus N ε φ is a F ε t -martingale and M ε φ t X uφ 2 du. b The proof is identical to a but with κ i for all i. We end this section by noting the trivial but very useful observation. Proposition For all non-negative measurable φ, a X ε φ X ε φ. b A ε φ Āε φ. Proof. a This is immediate from 2.3, and 2.4. b This is immediate from the definition of A ε and Āε. 29

36 2.3 Tightness Suppose ε n as n. The goal for this section is to establish the existence of continuous processes X, and A as weak limits points of {X εn } and {A εn }. We will do this by showing the two families of processes are C-relatively compact in DM F. Definition Let S, d be a Polish space. A collection of processes {X α } α I with paths in DS is C-relatively compact in DS iff it is relatively compact in DS and all weak limit points are continuous a.s. Definition D Cb R is separating iff for all µ, µ 2 M F, with µ φ µ 2 φ for all φ D, then µ µ 2. Theorem Jakubowski. Perkins, 22, II.4 Let D Cb be separating. A sequence of processes X n in DM F is C-relatively compact in DM F iff i φ D, the sequence {X n φ} n is C-relatively compact in DR. ii The compact containment condition holds, i.e, η, T > there is a compact set K η,t R such that sup P n sup Xt n K η,t > η t T η. Theorem Let ε n, as n, then a X εn is C-relatively compact in DM F. b A εn is C-relatively compact in DM F. Remark An immediate consequence of Theorem is that there exists a common subsequence ε n, and continuous M F -valued processes X and A such that X εn, A εn converges weakly to X, A. To prove Theorem 2.3.4, we will invoke Jakubowski s theorem using the separating class D Cc, smooths functions with compact support. We will prove both condition s i and ii of Theorem in the next 2 subsections. 3

37 2.3. Proof of Theorem 2.3.4: C-relative compactness of {X εn φ}, and {A εn φ} Our goal in this section is to show that for all φ C c, {X ε φ} and {A ε φ} are C- relative compactness in DR. Our main weapons to tackle this problem will the following theorems: Theorem Let {Y n } be a a family of stochastic processes with sample paths in CR. Suppose for all T >, there exists α, β, K > such that for all < s, t < T and n, E Y n t Then {Y n } is C-relatively compact in CR. Y n s α K t s +β. Proof. This is a well known result, and can be deduced from Corollary 3.8. and 3..3 in Ethier and Kurtz, 29. Theorem Slutsky s. Let S, d be a metric space, and X n, X be random elements of S S. If Y n Y and dx n, Y n weakly, then X n Y weakly. Proof. This is Theorem 3. in Billingsley, 23,. Recall that by 2.9 in Theorem 2.2.2, X ε t φ L ε tφ + M ε t φ + A ε tφ. We begin by simplifying A ε tφ. A ε tφ φ, φ x ε u ε s x Ξ ε ds, dx φ, φ x ε u ε s x Λ ε ds, dx + φ, φ x ε ε ψxuε s x dxds + φ, φ x ε u ε s x ˆΞ ε ds, dx φ, φ x ε u ε s x ˆΞ ε ds, dx I ε t φ + Âε tφ

38 Where I ε φ is continuous process R-valued process, and Âε φ is a Ft ε -martingale with jumps. Therefore Xt ε can be decomposed as Xt ε φ L ε tφ + Mt ε φ + It ε φ + Âε tφ. 2.3 To show the C-relative compactness of {X ε φ}, we will handle each term of 2.3 individually. Since L ε φ, A ε φ, and I ε φ all have sample paths in CR, Theorem tells us that we want to find estimate on the moments of the increments. Even though  ε φ contains jumps, we will show that it converges to in probability as ε thus allowing us to use Slutsky s theorem. We proceed to this in the next few lemmas. Before we can bound the increments of L ε φ, M ε φ, and I ε φ, the following estimate on the total mass of Xε will be useful. Lemma For all T >, and p 2 n, n, there are constants Cp, T such that for all t T, a E X t ε t, b E Xε t p Cp, T t p + δ p,t ε, Where δ p,t ε as ε. Proof. a By letting φ in 2., we get the that X ε solves the SDE X t ε M t ε + Āε t M t ε +, φ x ε Ξ ε ds, dx M t ε + εˆξ ε ds, dx + M t ε + εˆξ ε t + ε ε ψxdxds ελ ε ds, dx M t ε + εˆξ ε t + t. 2.3 Since M ε and ˆΞ ε are Ft ε -martingales, by taking expectations in 2.3 immediately gives us, E Xt t. 32

39 b Let T >. We now proceed by induction on n for p 2 n. In the case where p was shown in part a. Suppose that there is a Cp, T, δ p,t ε for ε such that, E Xt p Cp, T t p + δ p,t ε Lets now estimate the 2p-th moment. By applying Jensen s inequality for sums to 2.3 we get, E Xε t 2p E M t ε + εˆξ ε t + t 2p 3 2p E M ε t 2p + ε 2p ˆΞ ε t 2p + t 2p [ 3 2p E ε M t 2p + ε 2p E ˆΞ ε t 2p + t 2p] 3 2p C p [E [ M ε ] p t + ε 2p E [ˆΞ ε ] p t + + t 2p] p 3 2p C p [E X sds ε + ε 2p E Ξ εt ] p + ε 2p + t 2p The second last line is true for some C p > by the continuous version Burkholder- Davis-Gundy inequality for previsible jump martingales as stated in Perkins, 22, II.4, and by noting ˆΞ ε has jumps of size. The last line is true because [ M ε ] t M ε t X ε sds and ˆΞ ε t Ξ ε t. We will handle the two expectation terms in 2.33 individually. We bound the integral term using Jensen s inequality for x p and our induction hypothesis 2.32, E p X sds ε t p E t p X s ε p ds E Xε s p ds t p Cp, T s p + δ p,t εds Cp, T p + t2p + t p δ p,t ε Cp, T p + t2p + T p δ p,t ε

40 For the Ξ ε term in 2.33, we note that Ξ ε t Poi t ε. It is an elementary fact that the m-th moment of a Poisson random variable with mean λ is a polynomial in λ of degree m with zero constant term. Which implies E Ξ ε t p is a polynomial in t ε of degree p and thus, lim ε ε2p E Ξ ε t p Therefore combining 2.33, 2.34, and 2.35, we can conclude, E Xε t 2p [ ] Cp, T 3 2p C p p + t2p + T p δ p,t ε + ε 2p E Ξ ε t p + ε 2p + t 2p Cp, T 3 2p C p p + + t 2p + 3 2p C p T p δ p,t ε + ε 2p E Ξ ε t p + ε 2p C2p, T t 2p + δ 2p,T ε. Where δ 2p,T ε as ε. We will now use Lemma to estimate the increment moments of L εn φ, M εn φ and I εn φ. Lemma Let ε n as n, T >, φ C c exists K L p, T, φ, K M p, T, φ such that for all < s < t T, a E L εn t φ L εn s φ p K L p, T, φ t s p b E M εn t c E I εn t φ Ms εn φ 2p K M p, T, φ t s p φ Is εn φ p φ p t s p, and p 2 m for m. There 34

41 Proof. a We use the definition of L ε φ and applying Jensen s inequality for x p to get, E L εn t φ L εn s φ p t φ p E Xu εn du s 2 φ p p 2 E X u εn du s φ p 2 t s p E X u εn p du s φ p 2 t s p E Xε n u p du. We can esimate the integrand using Lemma s E L εn t φ L εn s φ p φ p 2 t s p Cp, T u p + δ p,t ε n du s φ p Cp, T 2 t s p p + t s p+ + δ p,t ε n t s φ p Cp, T 2 p + T p + δ p,t ε n t s p φ p Cp, T 2 p + T p + D p,t t s p. Since δ p,t ε n as n, δ p,t ε n is bounded by some D p,t uniformly in n. Therefore we have a constant K L p, T, φ such that for all n, E L εn t φ L εn s φ p K L p, T, φ t s p. b Note that by the definition of M ε φ in 2.8, Nt ε φ Mt ε φ Ms ε φ is a continuous Ft ε -martingale in t for t s with square function, N ε φ t s X ε uφ 2 du. Let s bound the p-th moments by applying Jensen s inequality for x 2p, and the 35

42 Burkholder-Davis-Gundy inequality. E Mt εn φ Ms εn φ 2p E Nt εn φ 2p C p E N εn φ p t p C p E Xu εn φ 2 du s p C p φ 2p E X u εn du s C p φ 2p t s p E X u εn p du C p φ 2p t s p Again, we apply Lemma to get, E Mt εn φ Ms εn φ 2p s s E Xε n u p du. C p φ 2p t s p C p, T u p + δ p,t ε n du s C p, T C p φ 2p t s p p + t s p+ + D p,t t s C p, T C p φ 2p p + T p + D p,t t s p. Just as in part a, we assumed δ p,t ε n is bounded by D p,t. Therefore there is a K M p, T, φ such that for all n, E Mt εn φ Ms εn φ 2p K M p, T, φ t s p. c We proceed directly by applying Hölder s inequality, t E It εn φ Is εn φ p E φ, φ x ε n p ψxu εn s s ε x dxds n p φ φ x ε n ψxdxds ε n s φ p t s p. 36

43 Lemma For all T >, φ C b. Âε tφ converges to uniformly on [, T ] in L 2 as ε. Proof. Since Âεn φ is a Ft εn -martingale, Doob s strong L 2 inequality tells us, E sup  ε tφ 2 4E Âε T φ 2 t T T 4E T 4 4 T 4T φ 2 ε, φ, φ x ε 2 u ε s x Ξ ε ds, dx φ, φ x ε 2 u ε s x Λ ε ds, dx φ 2 φ x ε 2 ε ψxdxds as ε. Therefore A εn φ converges to uniformly on [, T ] in L 2. With Lemma and 2.3. in hand, we are now ready to show {X εn φ} and {A εn φ} are C-relatively compact in DR. Proposition Suppose ε n, for all φ C c R, then a {X εn φ} is C-relatively compact in DR, b {A εn φ} is C-relatively compact in DR. Proof. a Let φ C b, and T >. Since X εn φ satisfies 2.3, X εn t φ L εn t Y εn t φ + M εn t φ + Âεn t φ. φ + I εn t φ + Âεn t φ Where Y εn φ is a stochastic process with sample paths in CR, and Âεn φ is a Ft εn -martingale with jumps. We will first show that Y εn φ is C-relatively compact. 37

44 For < s < t T, we proceed by bounding E Y εn t inequality to x 2p and Lemma E Y εn t Ys εn 2p E L εn t φ L εn s φ + M εn 3 2p E L εn t φ L εn t φ M εn s s φ 2p + E M εn t φ + I εn t φ M εn s Ys εn 2p, by applying Jensen s φ Is εn φ 2p φ 2p + E It εn φ Is εn φ 2p 3 2p K L 2p, T, φ t s 2p + K M p, T, φ t s p + φ 2p t s 2p 3 2p K L 2p, T, φt p + K M p, T, φ + φ 2p T p t s p K Y p, T, φ t s p, for some K Y p, T, φ. Therefore we have by Theorem 2.3.6, Y εn φ is C-relatively compact in CR. Suppose Y εn φ converges to some continuous weak limit point Y φ, by possibly passing through a subsequence. If d T represents the metric that induces the Skorohod J topology on D[, T ], then d T X εn φ, Y εn φ sup t T X εn t φ Y εn t φ sup  εn φ t T By Lemma 2.3. we have sup t T  εn φ converges to in L 2 and hence weakly. Therefore Slutsky s theorem tells us X εn φ Y φ weakly on [, T ]. Thus X εn is C-relatively compact in DR. b Theorem and Lemma 2.3.9c tell us that I εn φ is C-relatively compact in CR. It follows that A εn is C-relatively compact in DR, by identical reasoning as the last paragraph in the proof of a by replacing Y εn with I εn and X ε with A εn Proof of theorem 2.3.4: Compact containment condition Now that we have shown part i of Jakubowski s theorem, it remains to show that {X ε } and {A εn } satisfy the compact containment condition. Before we begin, it will be useful 38

45 to find an estimate on E Xε [ R, R] c for R large. For the remainder of this section, we fix φ R to be a a smooth approximation to [ R, R] c such that R + < x φ R x R < x. Lemma For all η, T > there exists a compact K η,t large enough such that for all t T E Xε t Kη,T c < η, uniformly in ε. Proof. Since X ε is a solution to 2., we have a Green s function representation of 2. which can be deduced from Perkins, 22, II.5, X ε t φ R P t s φ R Ā ε ds, dx + P t s φ R x Mds, dx, 2.36 Where P t is the Brownian semigroup. So we have E Xε t [ R, R + ] c E Xε t φ R E E P t s φ R Ā ε ds, dx P t s φ R, φ x ε Ξ ε ds, dx P t s φ R, φ x ε Λ ε ds, dx P t s φ R, φ x ε ψxdxds ε 39

46 We now need to control P t s φ R. If x R 2, P t φ R x e x y 2 2t φ R ydy 2πt y R tz+x R e x y 2 2t dy 2πt t z + x R z R x t z R 2 T 2π e z2 2 dz 2π e z2 2 dz 2π e z2 2 dz 2π e z2 2 dz. Where we made the substitution z y x t. By picking R large enough then we have for x R 2, PtφRx η 2T Also since ψ is integrable, we can pick R large enough such that We now return to 2.37, P t s φ R, φ x ε ε ψxdxds x R 2 P t s φ R, φ x ε ε ψxdxds + x R 2 ψxdx η 2T x < R 2 P t s φ R, φ x ε ε ψxdxds I + I

47 We will handle both I and I 2 separately. I t η 2T P t s φ R, φ x ε x R 2 ε ψxdxds P t s φ R φ x ε x R 2 ε ψxdxds φ R ε x R 2 ε ψxdxdt ψxdxds x R 2 η The first inequality was by Hölder, and the second was because P is a contractive semigroup. The third inequality was by Before we move onto I 2, we should note that for all bounded function f on R, x+ ε ε f, φx ε x fy φx εy dy ε ε φ x sup [x ε,x+ f ε y dy ε] ε sup [x ε,x+ f ε] sup f [x,x+] The last inequality was because we can assume ε. Therefore by combining 2.38, and 4

48 2.42 we get, I 2 t η 2T P t s φ R, φ x ε x < R 2 ε ψxdxdt x < R 2 x < R 2 sup P t s φ R ψxdxds [x,x+] η 2T ψxdxds η Finally, combining 2.37, 2.4, 2.4, and 2.43, we get the existence of a K η,t [R, R + ] such that, for all t T, E Xε t K c η,t η. Before proceeding to the compact containment condition for {X ε }, we will need the following estimates on the mass of L ε, M ε, and A ε. Lemma Suppose η, T, R >. If then we have the following: a E sup L ε tφ R φ R t T 2 T η, b E c E sup Mt ε φ R t T sup A ε tφ R t T 2 T η, T x >R sup E Xε t [ R, R] c < η, t T ψxdx. 42

49 Proof. a We proceed by directly using the definition of L ε. E sup L ε tφ R E sup Xu ε φr du t T t T 2 φ R T 2 E X u ε [ R, R] c du φ R T 2 E Xε u [ R, R] c du φ R 2 T η. b By Hölder s inequality and Doob s strong L 2 inequality we get, E sup Mt ε φ R t T E 2 sup Mt ε φ R 2 t T 2E M ε T φ R 2 2 2E M ε φ R T 2 T 2E T 2 Xsφ ε 2 2 Rds E Xsφ ε 2 R 2 ds T 2 E Xε s [ R, R] c ds 2 T η. 2 43

50 c Finally, E sup A ε tφ R t T E E sup Ā ε tφ R t T sup t T T E T T φ R, φ x ε Ξ ε ds, dx φ R, φ x ε Ξ ε ds, dx φ R, φ x ε Λ ε ds, dx φ R, φ x ε ε ψxdxds T φ R, φ x ε ε ψxdx. Since we are taking limits as ε, we can assume ε <. Therefore by 2.42 we have, ε φ R, φ x ε sup y x + φ R y ε sup R < x R < x. y x + So we have shown, E sup A ε tφ R t T T x R ψxdx. We now prove that {X ε } and {A ε } satisfy the compact containment condition. Proposition {X ε } and {A ε } satisfies the compact containment condition. I.e, For all η, T >, there is a compact K η,t such that, a sup P ε b sup P ε sup Xt ε Kη,T c > η t T sup A ε tkη,t c > η t T < η, < η. 44

51 Proof. a Let η, T >. By Lemma we can find R large enough, depending only on η and T such that, sup E Xε t [ R, R] c < η. t T Where η > will be decided later. By using R + < x φ R x and Lemma 2.3.3, we have P sup X ε [ R, R + ] c > η t T P sup X ε φ R > η t T E sup t T X ε φ R [ η E sup L ε η tφ R + E sup Mt ε φ R t T t T φ K η 2 T η + 2 T η + T x >R + E Since ψ is integrable, we can pick R large enough so that x >R sup A ε tφ R t T ψxdx ] ψxdx < η Therefore by letting η be small enough, such that 2.44 is less than η. Therefore we have found a compact set K η,t [ R, R + ], such that, independent of ε. P sup X ε Kη,T c > η t T < η, 2.46 b Let R and K η,t be as above. By using x > R + φ R x and Lemma 2.3.3, we 45

52 have P sup A ε Kη,T c > η t T P sup A ε φ R > η t T E sup t T A ε φ R T η η x >R ψxdx 2.47 < η, independent of ε. The last inequality was because 2.47 was smaller than 2.44 which is less than η. Thus we have finally proven Theorem SPDE characterization of limit points We now have by Theorem the existence of a sequence ε n as n and continuous M F -valued processes X, A such that X εn, A εn converge weakly to X, A, possibly by taking subsequences. Let F t be the filtration generated by the processes X and A, ie., F t σ X s, A s s t. We want to show that for all φ C b, X admits an SPDE representation of the form, X t φ L t φ + M t φ + A t φ. Where L t φ φ X s ds, 2 and Mφ is a continuous F t -martingale with quadratic variation Mφ t X sφ 2 ds. By rearranging the terms in 2.9, we have for all φ C b, M εn t φ X εn t φ L εn t φ A εn t φ,

53 is a Ft εn -martingale with quadratic variation M εn t to argue we get our desired Mφ by taking limits of M εn Xεn s φ 2 ds. The plan of attack as n. Before we proceed, it will be convenient to work with almost sure convergence as opposed to weak. Theorem 2.4. Skorohod Representation Theorem. Billingsley, 23, p. 7 Let S, d be a Polish space, and suppose X n, X, be S-valued random variables on possibly different probability spaces such that X n converges weakly to X. There exists a probability space Ω, F, P and random variables X n, X with the same law as X n, X and P-a.s, X n ω S,d n Xω. For the remainder of this paper we will assume that without loss of generality that we are working on a common probability space where X εn, A εn converges a.s. to X, A in DM F DM F. Note that for all φ C b and X X s φ C b, a.s. φ 2, X Xφ in a continuous map from DM F DR ds is a continuous map from DM F CR. This implies for all lim n Xεn t lim n Aεn t φ X t φ, φ A t φ, and, lim n Lεn t φ lim n X εn s φ 2 ds φ X s ds L t φ. 2 Therefore a.s., lim M εn n t φ lim n Xεn t φ L εn t X t φ L t φ A t φ φ A εn t φ M t φ 2.49 If N εn φ M εn φ 2 M εn φ is the square Ft εn -martingale for M εn φ. Similar to above, for all φ C b, X X sφ 2 ds is a continuous map from DM F CR, 47

54 which implies a.s. lim N εn n t φ lim M εn n t lim M εn n t M t φ 2 φ 2 M εn φ t φ 2 X s φ 2 ds Xs εn φ 2 ds N t φ. 2.5 Since X, L and A are continuous, we have that Mφ and Nφ are also continuous. It remains to show that Mφ and Nφ are F t -martingales. Note that for all T >, Burkholder-Davis-Gundy inequality and Jensen s inequality for x 2 gives us, E sup Mt εn φ 4 CE M εn φ 2 T t T T CE CT E T CT φ 4 E CT φ 4 2 Xs εn φ 2 ds Xs εn φ 2 2 ds T T X s εn 2 ds We can estimate the integral in 2.5 by applying Lemma 2.3.8, E sup Mt εn φ 4 CT φ 4 t T T E Xε n s 2 ds 2.5 C2, T s 2 + δ 2,T ε n ds C2, T T 3 + T δ 2,T ε n CT φ 4 3 C2, T T C φ T D 2,T 3 < The last line used the fact that δ 2,T ε n as n, and is thus uniformly bounded in n by some D 2,T. 48

55 Hence we have shown that Mt εn φ and Nt εn φ are L 4 and L 2 bounded respectively uniformly in n and t T. Therefore {sup t T M εn t φ} and {sup t T Nt εn φ} are uniformly integrable, which along with 2.49 and 2.5 imply M εn φ Mφ and N εn φ Nφ in L. L convergence and the fact that M εn φ and N εn φ are Ft εn -martingales, imply for all Ψ : M 2m F R be bounded and continuous and s,... s m, s t, lim E M εn n t εn E M lim n t φ Ms εn φψxs εn,... Xs εn m, A εn s,..., A εn s m φ Ms εn φψxs εn,... Xs εn m, A εn s,..., A εn s m E M t φ M s φψx s,... X sm, A s,..., A sm Similarly, lim E N εn n t εn E N lim n t φ Ns εn φψxs εn,... Xs εn m, A εn s,..., A εn s m φ Ns εn φψxs εn,... Xs εn m, A εn s,..., A εn s m E N t φ N s φψx s,... X sm, A s,..., A sm Therefore we can conclude by 2.52, and 2.53 that Mφ and Nφ are F t -martingales. Since N t φ M t φ 2 X sφ 2 ds is a F t -martingale, we have by the uniqueness of quadratic variation, Mφ t X sφ 2 ds. We have proven the following. Theorem If X, A is a weak limit point of X εn, A εn as ε n, then for all φ C b R, X satisfies the SPDE, X t φ L t φ + M t φ + A t φ Where Mφ is a continuous F t -martingale with quadratic variation, 2.5 Density of X Mφ t X s φ 2 ds. We now have the existence of a sequence ε n as n, and continuous M F -valued processes X, that satisfies the SPDE 2.54, such that X εn converges weakly to X. 49