On Markov Chain Approximations to Semilinear Partial Differential Equations Driven by Poisson Measure Noise

Transcription

1 MARCEL DEKKER, INC. 7 MADISON AVENUE NEW YORK, NY 6 3 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc. STOCHASTIC ANALYSIS AND APPLICATIONS Vol., No., pp , 3 On Markov Chain Approximations to Semilinear Partial Differential Equations Driven by Poisson Measure Noise Michael A. Kouritzin, Hongwei Long,* and Wei Sun Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Canada ABSTRACT We consider the stochastic model of water pollution, which mathematically can be written with a stochastic partial differential equation driven by Poisson measure noise. We use a stochastic particle Markov chain method to produce an implementable approximate solution. Our main result is the annealed law of large numbers establishing convergence in probability of our Markov chains to the solution of the stochastic reaction-diffusion equation while considering the Poisson source as a random medium for the Markov chains. Key Words: Stochastic reaction diffusion equations; Markov chains; Poisson processes; Annealed law of large numbers. *Correspondence: Hongwei Long, Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Canada T6G G; Fax: ; long@math.ualberta.ca. 49 DOI:.8/SAP-993 Copyright q 3 by Marcel Dekker, Inc (Print); (Online)

2 MARCEL DEKKER, INC. 7 MADISON AVENUE NEW YORK, NY 6 3 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc. 4 Kouritzin, Long, and Sun. INTRODUCTION AND NOTATION Based upon Kallianpur and Xiong, [7] we consider a stochastic pollution model which characterizes the transport of contaminants (e.g. chemical or bacterial) in a moving sheet of water. Suppose that there are r sources of contamination located at different sites k,...,k r in the water region E ¼ ½; L Š ½; L Š: The r sources disperse contaminants at the jump times of independent Poisson processes N (t),, N r (t). The magnitude of the j th contaminant released by the i th source is A j i ; where {Aj i ; j ¼ ; ;...; i ¼ ;...; r} are independent random variables that are independent of {N,..., N r }, and {A j i ; j ¼ ; ;...} has common distribution F i(da). The contaminants are initially distributed in the area Bðk i ; Þ ¼{x : jx k i j, }, ð; L Þ ð; L Þ according to a proportional function u i (x) satisfying u i ðxþ $ and u i ðxþdx ¼ : Bðk i ;Þ Each u i is continuous and zero off B(k i,). Upon release, the contaminants diffuse and drift in the water sheet. Also, there is the possibility of nonlinear reaction of the contaminants due to births and deaths of bacteria or chemical adsorption, which refers to adherence of a substance to the surface of the porous medium in groundwater systems. Reaction is modeled by nonlinear term R(u) below. The stochastic model described above can be written mathematically as follows (abbreviating U x ; U x ; D U þ ; and 7 U ð Þ T Þ uðt; xþ ¼DDuðt; xþ V 7uðt; xþþrðuðt; xþþ t subject to þ Xr X j¼ A j i ðvþu iðxþ t¼t j i ðvþ; x [ ½; L Š ½; L Š; ð:þ uðt; L ; x Þ¼ uðt; ; x Þ¼; uðt; x ; L Þ¼ uðt; x ; Þ ¼; uð; xþ ¼u ðxþ; where u(t,x) denotes the concentration of a dissolved or suspended substance, D. denotes the dispersion coefficient, V ¼ðV ; V Þ with V. ; V ¼ denotes the water velocity, R( ) denotes the nonlinear reaction term, {t j i ; j [ þ} are the jump times of the independent Poisson processes N i ðtþði ¼ ; ; ; rþ with parameters h i, and u (x) denotes

3 MARCEL DEKKER, INC. 7 MADISON AVENUE NEW YORK, NY 6 3 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc. Markov Chain Approximations 4 the initial concentration of the contaminants in the region ½; L Š ½; L Š: All the random variables A j i and t j i (or N i (t)) are defined on some probability space (V,F,P). Moreover, we assume that R : ½; Þ! R is continuous with RðÞ $ and sup RðuÞ, ; ð:þ u. þ u and for some q $ and K. ; we have the local Lipschitz condition jrðuþ RðvÞj # Kju vjð þjuj q þjvj q Þ; jrðuþj # Kð þ u q Þ: ð:3þ Let us define a differential operator A ¼ DD V 7 with Neumann boundary conditions in both variables. We take the initial domain D (A)ofA to be { f [ C ðeþ : f ð; x Þ¼ f ðl ; x Þ¼ f ðx ; Þ ¼ f ðx ; L Þ¼}; where C (E) denotes the twice continuously differentiable functions on E. Letting rðxþ ¼e cx and c ¼ V D A ¼ D rðxþ þ rðxþ x x ; we can rewrite A as x In the sequel, ðh;, ;.Þ is the Hilbert space L (E,r(x)dx). Then, (A,D (A)) is symmetric on H and admits a unique self-adjoint extension with domain DðAÞ ¼{f [ H : j7f j; Df [ H and f ð; x Þ¼ f ðl ; x Þ¼ ; f ðx ; Þ ¼ f ðx ; L Þ¼}: We define random process Q(t) by Qðt; x; vþ ¼ Xr : u i ðxþ XN iðtþ A j i ðvþ for t $ ; x [ ½; L Š ½; L Š; v [ V; j¼ and find that the equation (.) can be rewritten as duðt; xþ ¼½Auðt; xþþrðuðt; xþþšdt þ dqðt; xþ; uðþ ¼u : ð:4þ We consider mild solutions uðtþ ¼TðtÞu þ Tðt sþrðuðsþþds þ Tðt sþdqðsþ ð:5þ of our stochastic partial differential equation (SPDE) (.4), where T(t) is the C -semigroup generated by the operator A. For any separable Hilbert space V, C V [,T ] and D V [,T ] denote respectively the V-valued continuous and càdlàg functions h such that

4 MARCEL DEKKER, INC. 7 MADISON AVENUE NEW YORK, NY 6 3 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc. 4 Kouritzin, Long, and Sun hðtþ [ V for all # t # T: For càdlàg functions h, we define ( t ¼ ; hðt Þ U lim sbt hðsþ, t # T: We shall use the notations C,C(N,l),C(T) and so on, for finite constants (depending on N, l or T etc.), which may be different at various steps in the proofs of our results. In this note, we shall discuss unique D H [,T ]-valued solutions and Markov chain approximations to SPDE (.4). In Ref. [], Kouritzin and Long established the quenched law of large numbers for the Markov chain approximations to SPDE (.4) for each fixed path of the Poisson sources and gave an annealed law of large numbers, where the Poisson source is treated as a random medium, as a corollary. It turns out that a more general annealed result is possible if alternative methods are used. In this note, we shall use a different method to establish a general annealed law of large numbers for the Markov chain approximations of the stochastic reaction diffusion model. We remark that our hypotheses (to follow in section 3) are weaker than those given in Ref. [] In Ref. [], uniform boundedness was imposed on u. Here, we only require that the expectation of u is bounded. Also, there is significant difference between our current proof method and the one used in Ref. [] In Ref. [], a relative compactness method and Skorohod representation theorem were crucial in the proof of laws of large numbers. In this note, we directly apply Cauchy criterion (convergence in probability) to our Markov chains and utilize the nice regularity property of Green s function. The current method is clearer and more elegant, especially for people with stronger analysis vis-à-vis probability background. The contents of this note are organized as follows: In Section, we review the Markov chain approximations to our pollution model (.4) via the stochastic particle method and the random time changes approach. In Section 3, we show that there exists a unique mild solution to (.4) and prove the annealed law of large numbers.. CONSTRUCTION OF MARKOV CHAIN The Markov chain approximation discussed in this paper is motivated by the stochastic particle simulation method for differential equations studied by Kurtz, [] Arnold and Theodosopulu, [] Kotelenez, [8,9] Blount, [3 5] and Kouritzin and Long. [] They proved that a sequence of Markov chain approximations converges to the unique solution of a reaction-diffusion or

5 MARCEL DEKKER, INC. 7 MADISON AVENUE NEW YORK, NY 6 3 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc. Markov Chain Approximations 43 stochastic reaction diffusion equation in probability. Stochastic limits were considered in Ref. [] In this case, the Markov chain approximations have two kinds of randomness, which are the external fluctuation coming from the Poisson sources and the internal fluctuation in implementing the reaction and diffusion. Here, we state some necessary background and results from Ref. [] for our later development. We denote by {ðl p ; f p Þ} p¼ð p ; p Þ[ðN Þ the eigenvalues and eigenfunctions of A (see Lemma. of Kouritzin and Long [] for their expressions). Now, we introduce the discretized approximation operator A N of A. We divide ½; L Þ ½; L Þ into L N L N cells of size N N : I k U k N ; k N k ¼ ; ;...; L N: k N ; k ; k ¼ðk ; k Þ; k ¼ ; ;...; L N; N We also define the class of cells where the contaminants can enter K N i ¼ fk : I k, Bðk i ; Þg; i ¼ ; ; ; r: Let H N ¼ {w [ H : w is constant on each I k }. We define the following discrete gradients: and, 7 Nxi f ðxþ ¼N f x þ e i f x e i N N 7 þ Nx i f ðxþ ¼N f x þ e i f ðxþ N ; 7 Nx i f ðxþ ¼N f x e i f ðxþ ; i ¼ ; ; N where e ¼ð; Þ and e ¼ð; Þ: We define A N by A N f ðxþ U D r ~ 7 Nx ðr ~ 7 Nx ÞþD Nx f ðxþ; ð:þ

6 MARCEL DEKKER, INC. 7 MADISON AVENUE NEW YORK, NY 6 3 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc. 44 Kouritzin, Long, and Sun where D Nx f ðxþ ¼7 þ Nx ð7 Nx f ÞðxÞ ¼ N f x þ e f ðxþþf x e : N N In order to take the boundary conditions into account for the discretized approximation scheme, we extend all function f [ H N to the region [ N ; L þ N Š ½ N ; L þ NŠ by letting f ðx ; x Þ¼f x þ N ; x ; x [ N ; ; x [ ½; L Š; f ðx ; x Þ¼f x N ; x ; x [ L ; L þ ; x [ ½; L Š; N f ðx ; x Þ¼f x ; x þ ; x [ ½; L Š; x [ N N ; ; f ðx ; x Þ¼f x ; x ; x [ ½; L Š; x [ L ; L þ N N and denote this class of functions by H N bc : Then, HN bc is the domain of AN.We denote by {l N p ; fn p }ðl N;L NÞ p¼ð p ;p Þ¼ the eigenvalues and eigenfunctions of A N. For their precise expressions, we refer to p Lemma. of Kouritzin and Long. [] For p ¼ðp ; p Þ [ N ; let jpj ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p þ p : Let T N ðtþ ¼expðA N tþ: Then, f N p are eigenfunctions of T N (t) with eigenvalues exp{l N p t}: Now, we describe the stochastic particle systems. Let l ¼ lðnþ be a function such that lðnþ! as N!: l can loosely be thought of as the mass or the amount of concentration of one particle. We let n k (t) denote the number of particles in cell k at time t for k ¼ðk ; k Þ [ {;...; L N} {;...; L N} and also, to account for our Neumann boundary conditions, we set n ;k ðtþ ¼n ;k ðtþ; n L Nþ;k ðtþ ¼n L N;k ðtþ; k ¼ ;...; L N; n k ;ðtþ ¼n k ;ðtþ; n k ;L NþðtÞ ¼n k ;L NðtÞ; k ¼ ;...; L N: Particles undergo diffusion between adjacent cells, and give births or die in each cell due to reaction. For the transition rates of particles evolving in cells, we refer to Kouritzin and Long. [] Let (V, F, P ) be another

7 MARCEL DEKKER, INC. 7 MADISON AVENUE NEW YORK, NY 6 3 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc. Markov Chain Approximations 45 probability space on which is defined independent standard Poisson processes {X k;r þ;n ; Xk;R ; N ; Xk; þ;n ; Xk; ; N ; Xk; þ;n ; Xk; ; N }ðl N;L NÞ k¼ð;þ ; {X k; þ;n ; Xk; ; N ; k ¼ ð; k Þ} L N k ¼ and {X k; þ;n ; Xk; ; N ; k ¼ðk ; Þ} L N k ¼ : We define the product probability space ðv ; F ; P Þ¼ðV V; F ^ F; P PÞ: In the sequel, brc denotes the greatest integer not more than a real number r. For any real a, a þ U a _ and a U ða ^ Þ: Then, from Ethier and Kurtz [6] and Kouritzin and Long, [] we have for k [ {ð; Þ;...; ðl N; L NÞ} n N k ðtþ¼nn k ðþþxk;r þ;n l R þ ðn N k ðsþl Þds X k;r ;N l R ðn N k ðsþl Þds þ X X k;i ;N d i;n ðnn k ðsþþds X X ke i;i ;N þ Xr X j¼ d i;n ðnn ke i ðsþþds X k;i þ;n X ke i;i þ;n d þ i;n ðnn k ðsþþds d þ i;n ðnn ke i ðsþþds blu i ðkþa j i þ:5c t$t j i k[k N i ; ð:þ where d ;N ðn k Þ¼DN e Nn c kþe DN e Nn c k and d ;N ðn k Þ¼DN n kþe DN n k : Equation (.) is our Markov chain approximation to equation (.4) that can be implemented directly on a computer. We let {G N t } t$ be the smallest right continuous standard filtration such that {X k;r s;n ðlr t R s ðn N k ðsþl ÞdsÞ; X k;i s;n ð R t ds i;n ðnn k ðsþþdsþ; s ¼þ; ; i ¼ ; }ðl N;L NÞ k¼ð;þ ; {X k; s;n ðr t d ;N s ðnn k ðsþþdsþ; s ¼ þ; ; k ¼ð; k Þ} L N k ¼ ; and {Xk; s;n ðr t ds ;N ðnn k ðsþþdsþ; s ¼þ; ; k ¼ ðk ; Þ} L N k ¼ as well as{n i; i ¼ ; ;...; r} are adapted to {G N t }: To get the concentration in each cell, we divide n N k ðtþ by l u N ðt; xþ ¼ XL N X L N k ¼ k ¼ n N k ðtþ k ðxþ; ð:3þ l where k ( ) denotes the indicator function on I k. Then, from, (.) it follows

8 MARCEL DEKKER, INC. 7 MADISON AVENUE NEW YORK, NY 6 3 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc. 46 Kouritzin, Long, and Sun that where u N ðtþ ¼u N ðþþ Q N ðt; Þ ¼ Xr N ðtþ U X N iðtþ j¼ k[k N i ðl N;L X NÞ l k¼ð;þ A N u N ðsþds þ X l blu i ðkþa j i ðvþþ:5c kð Þ; N k;r;þ ðtþþn k;r; ðtþþx and N k;r;þðtþ ¼Xk;R þ;n l R þ ðn N k ðsþl Þds l Rðu N ðsþþds þ N ðtþþq N ðtþ; ð:4þ " # N k;i ðtþ N ke i ;i ðtþ k ; N k;r;ðtþ ¼Xk;R ; N l R ðn N k ðsþl Þds þ l N k;iðtþ ¼Xk;i þ;n d þ i;n ðnn k ðsþþds X k;i ; N d i;n ðn N k ðsþþds; i ¼ ; R þ ðn N k ðsþl Þds; R ðn N k ðsþl Þds; d i;n ðnn k ðsþþds are L - martingales with respect to {G N t } under probability measure P (see Lemma.5 of Kouritzin and Long [] ). By variation of constants and (.4), it follows that u N ðtþ ¼u N ðt; v Þ satisfies u N ðtþ ¼T N ðtþu N ðþþ T N ðt sþrðu N ðsþþds where Y N ðtþ ¼ þ Y N ðtþþ T N ðt sþd N ðsþ: T N ðt sþdq N ðsþ; ð:5þ

9 MARCEL DEKKER, INC. 7 MADISON AVENUE NEW YORK, NY 6 3 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc. Markov Chain Approximations LAW OF LARGE NUMBERS: ANNEALED APPROACH In this section, we shall prove the law of large numbers for u N via annealed approach, which means that N i (t) and A j i are considered to be random variables, and the Markov chain u N evolves in this random medium. As already mentioned in Section, u N is defined on the product probability space ðv ; F ; P Þ¼ðV; F ; PÞ ðv; F ; PÞ: For f : E! R; let k f k ¼ sup x[e j f ðxþj: In the sequel, we always consider the Skorohod metric d on D H [,T ] so that (D H [,T ],d) is a complete separable metric space. We introduce the following hypotheses (abbreviating E U E P Þ : Hypotheses (i) ke ðu N ðþ q k # C, : (ii) {l(n)} is any sequence satisfying lðnþ!as N!: (iii) ku N ðþ u k! in probability P. (iv) ku N ðþk # CðN; lþ, : (v) The R distribution of the deposit magnitudes satisfies R a q F þ i ðdaþ, : (vi) E ku k # c, : We have the following annealed law of large numbers: Theorem 3.. Let the Hypotheses be fulfilled. Then, there exists a pathwise unique mild solution u to (.4) and supku N ðtþ uðtþk! ð3:þ in probability P as N!. Remark 3.. Let {N k } k¼ be an increasing sequence in N such that N k! as k!: We define V ~ ¼ Q ~ m¼ V m ; where V ~ m ¼ D R L Nm L Nm ½; Þ: Set ~F ¼ ^ m¼ Bð V ~ m Þ; which is the s-algebra generated by open sets under Skorohod topology and countable products. From Lemma.5 of Kouritzin and Long, [] we know that n N ðtþ ¼{n N k ðtþ}ðl N;L NÞ k¼ð;þ is well defined, and for each v [ V there exists a unique probability measure P v on (Ṽ, ~F) such that Pð v [ V : n N m ð v; vþ [ A ;...n N m j ð v; vþ [ Aj Þ¼ P ~ v ð ~v [ V ~ : ~v m [ A ;...; ~v mj [ A j Þ for all A i [ BðD R L Nm i L Nm i ½; ÞÞ; i ¼ ;...; j; j ¼ ; ;...: Moreover, we have that for each B [ ~F; v! ~P v ðbþ is (V,F)-measurable, and

10 MARCEL DEKKER, INC. 7 MADISON AVENUE NEW YORK, NY 6 3 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc. 48 Kouritzin, Long, and Sun v! R ~ V f ðv; ~vþ ~P v ðd ~vþ is F-measurable for each bounded measurable function f. We can write P ðdv Þ¼ ~ P v ðd ~vþpðdvþ; v ¼ðv; ~vþ: The statements of Lemma 3.3 and Lemma 3.4 of Ref. [] are still valid if we replace Ẽ v by E. We may just take expectations with respect to P. Lemma 3.3. For b [ ½; qš; supke ðu N ðsþþ b k # Cðt; l; ke ðu N ðþþ b k Þ, ; s # t where C is decreasing in l. Proof. Our proof follows the proof of Lemma 3.5 of Ref. []. The only significant modification is to estimate the expectation (under P ) of the last term on the right-hand side of (3.) in the proof of Lemma 3.5 of Ref. [].In fact, letting s N ðkþ ¼ R I k rðxþdx and using (3.) in the proof of Lemma 3.5 of Ref. [], we have b E T N ðt sþdq N ðsþ; ðs N ðkþþ k X r X N iðtþ b # E ku i k A j i þl b E jn ðtþþ þ N r ðtþj b A: j¼ ð3:þ Recalling that {N i, i ¼,,...,r} are independent Poisson processes with parameters h i, we set N i ð½; tš BÞ ¼ XN iðtþ B ða j i Þ; j¼ which is a Poisson measure with characteristic measure m i ðbþ ¼h i F i ðbþ: Denote by N ~ i ðdsdaþ the compensated martingale measure of N i (dsda). Wald s equation gives " # X N iðtþ E A j i ¼ th i af i ðdaþ ¼th i a i ; a i U af i ðdaþ: ð3:3þ R þ R þ j¼

11 MARCEL DEKKER, INC. 7 MADISON AVENUE NEW YORK, NY 6 3 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc. Markov Chain Approximations 49 Applying Jensen s inequality and Burkholder s inequality, we find that X r X N iðtþ b E ku i k A j i # r bxr # r bxr # r bxr j¼ ku i k b E X N b iðtþ A j i j¼ ku i k b E th i a i þ a ~ N i ðdsdaþ R þ " # ku i k b b ðth i a i Þ b þ E a N ~ b i ðdsdaþ R þ ( ) b # ðrþ bxr ku i k b t b h b i a b i þ CðbÞE a N ~ i ðdsdaþ : R þ When b # ; we have by Jensen s inequality E R þ a ~ N i ðdsdaþ b t b t ð3:4þ b # th i a R F i ðdaþ ; ð3:5þ þ and otherwise (when b. ) we have by Jensen s inequality that " # b E a N ~ X N b iðtþ i ðdsdaþ ¼ E ða j i Þ R þ t j¼! 3 ¼ X X N b iðtþ E ða j 4 i Þ N iðtþ¼n5p ðn i ðtþ¼nþ n¼ # X n¼ j¼ " # X n n b E ða j i Þb N i ðtþ¼n P ðn i ðtþ¼nþ j¼ ¼ E ðn i ðtþþ b a b F i ðdaþ: Rþ ð3:6þ

12 MARCEL DEKKER, INC. 7 MADISON AVENUE NEW YORK, NY 6 3 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc. 43 Kouritzin, Long, and Sun Thus, by (3.4), (3.5), (3.6) and Hypothesis (v), one has that X r X N iðtþ b E ku i k A j i # CðtÞ: j¼ Therefore, noting E jn ðtþþ þn r ðtþj b # CðtÞ; we have by (3.) that D E b E T N ðt sþdq N ðsþ;ðs N ðkþþ k # Cðt;lÞ, : Now, the lemma follows from the proof of Lemma 3.5 of Ref. [] A Lemma 3.4. sup ky N ðtþk! in probability P as N!: Proof. Note that Lemma 3.6 of Kouritzin and Long [] is still valid under our new Hypotheses if we replace P ~ v by P and E ~ v by E. Since the proof is almost the same as that of Lemma 3.6 (iv) of Ref. [], we omit the details here. Immediately from Lemma 3.7 of Ref. [], we have the following lemma. Lemma 3.5. For almost all v [ V sup T N ðt sþdq N ðsþ Tðt sþdqðsþ! as N!: The remainder of the development differs substantially from Ref. [] Lemma 3.6. There exists a càdlàg H-valued process w such that sup ku N ðtþ wðtþk! in probability P as N!: Proof. We apply Cauchy criterion to prove our statement. It follows from (.5) that for N; M [ N ku N ðtþ u M ðtþk #kt N ðtþu N ðþ T M ðtþu M ðþk þ ky N ðtþk þ ky M ðtþk þ T N ðt sþdq N ðsþ T M ðt sþdq M ðsþ þ þ T N ðt sþrðu N ðsþþds T M ðt sþrðu M ðsþþds Tðt sþrðu N ðsþþds Tðt sþrðu M ðsþþds

13 MARCEL DEKKER, INC. 7 MADISON AVENUE NEW YORK, NY 6 3 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc. Markov Chain Approximations 43 þ ktðt sþ Rðu N ðsþþ Rðu M ðsþþ kds U V N;M t þ ktðt sþðrðu N ðsþþ Rðu M ðsþþþkds: ð3:7þ We claim that sup V N;M t! in probability as N; M!(in any way). This is true by Hypothesis (iii), Trotter-Kato theorem, Lemma 3.4, and Lemma 3.5 if we can prove that sup T N ðt sþrðu N ðsþþds Tðt sþrðu N ðsþþds converges to zero in probability when N!: Note that T N ðt sþrðu N ðsþþds ¼ jpj#n Tðt sþrðu N ðsþþds X expðl N p ðt sþþkfn p ; Rðu N ðsþþldsf N p expðl p ðt sþþ X kf p ; Rðu N t ðsþþldsf p þ expðl N p ðt sþþkfn p ; Rðu N ðsþþldsf N p jpj.n X expðl p ðt sþþkf p ; Rðu N ðsþþldsf p : jpj.n ð3:8þ For jpj ; c jl p j # jl N p j # c jl p j and jl p j $ c 3 jpj (see Lemma. and Remark.3 of Kouritzin and Long [] ). So, by Cauchy-Schwarz inequality, we have expðl N p ðt sþþkfn p ; Rðu N ðsþþlds # expðl N p ðt sþþds # Cjpj kf N p ; Rðu N ðsþþl ds: kf N p ; Rðu N ðsþþl ds

14 MARCEL DEKKER, INC. 7 MADISON AVENUE NEW YORK, NY 6 3 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc. 43 Kouritzin, Long, and Sun Thus, X expðl N p ðt sþþkfn p ; Rðu N ðsþþlds jpj.n X T # Cn kf N p ; Rðu N ðsþþl ds # Cn k; R ðu N ðsþþlds: p It follows by (.3), Hypothesis (i) and Lemma 3.3 that X 3 t E sup expðl N p ðt sþþkfn p ; Rðu 4 N ðsþþldsf N 5 p # CðTÞ n : jpj.n Similarly, we can show that X 3 t E 4sup expðl p ðt sþþkf p ; Rðu N ðsþþldsf p 5 # CðTÞ n : jpj.n Note that for fixed p, jl N p l pjþkf N p f pk! as N! and sup p;n ðkf N p k þkf p k Þ, (see Lemma., Lemma. and Remark.3 of Ref. [] ) Therefore, by Lemma 3.3, it follows that for p fixed, sup expðl N p ðt sþþkfn p ; Rðu N ðsþþldsf N p expðl p ðt sþþkf p ; Rðu N ðsþþldsf p converges to zero in probability as N!: Thus, sup T N ðt sþrðu N ðsþþds Tðt sþrðu N ðsþþds! ð3:9þ in probability as N!: Next, we come back to the inequality (3.7). We remark that the semigroup T(t) is defined by ðtðtþf ÞðxÞ ¼ R E Gðt; x; yþf ðyþrðyþdy; x [ E; where G(t;x,y) is the Green s function corresponding to A and is given by X Gðt; x; yþ ¼ expðlp tþf p ðxþf p ðyþ; t $ : p

15 MARCEL DEKKER, INC. 7 MADISON AVENUE NEW YORK, NY 6 3 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc. Markov Chain Approximations 433 Moreover, using Lemma. of Ref. [], we find that for any t [ [,T ], there exists a constant cðtþ. such that E ð;þ jgðt; x; yþj rðxþdx ¼ X e lpt f p ðyþ p¼ð;þ # C X ¼ C e l p t X e l p t p ¼ p ¼ þ X p ¼ þ X # C þ þ p ¼ exp Dp! p þ Dc t L exp Dp! p t L exp Dp tx dx L exp Dp tx dx ¼ C þ L p ffiffiffiffiffiffiffiffiffiffiffi þ L p Dpt ffiffiffiffiffiffiffiffiffiffiffi # cðtþ t : Dpt By using Minkowski s integral inequality, (.3), and Cauchy-Schwarz inequality, we find that Tðt sþðrðu N ðsþþ Rðu M ðsþþþ ( ) ¼ Gðt s; x; yþðrðu N ðs; yþþ Rðu M ðs; yþþþrðyþdy rðxþdx # E E E E # CðTÞðt sþ jgðt s; x; yþj rðxþdx E L jrðu N ðs; yþþ Rðu M ðs; yþþjrðyþdy ju N ðs; yþ u M ðs; yþjð þju N ðs; yþj q

16 MARCEL DEKKER, INC. 7 MADISON AVENUE NEW YORK, NY 6 3 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc. 434 Kouritzin, Long, and Sun þju M ðs; yþj q ÞrðyÞdy # CðTÞðt sþ ku N ðsþ u M ðsþk ð þju N ðs; yþj q þju M ðs; yþj q Þ rðyþdyo : ð3:þ For convenience, let : f ðsþ ¼ ð þju N ðs; yþj q þju M ðs; yþj q Þ rðyþdy E The following method is motivated in part by Kouritzin. [] Substituting (3.) into (3.7), we get ku N ðtþ u M ðtþk # V N;M t þ CðTÞ By Lemma 3.3, it is easy to see that for t [ [,T ] E ðt sþ f ðsþds # CðTÞ, and consequently ðt sþ f ðsþds, ; a:s: By iterating (3.) n times, we get ku N ðtþ u M ðtþk # supv N;M s þ CðTÞ s#t s s sn sn E ðt sþ f ðsþku N ðsþ u M ðsþkds a:s: ð3:þ f ðsþ pffiffiffiffiffiffiffiffiffiffi ds þ þðcðtþþ n t s f ðsþ f ðs n Þ p ffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t s s n s n f ðsþ f ðs n Þ p ffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t s s n s n ds n ds ds þðcðtþþ nþ ku N ðs n Þ u M ðs n Þkds n ds ds; ð3:þ If q ¼ ; then f ðsþ # C, : This case is much simpler than the case of

17 MARCEL DEKKER, INC. 7 MADISON AVENUE NEW YORK, NY 6 3 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc. Markov Chain Approximations 435 q. : So, we only deal with the later case here. For q., by applying Hölder s inequality and Young s inequality, we find f ðsþ pffiffiffiffiffiffiffiffiffiffi ds # t s h ðt sþ i q qþ ds qþ q f q q ðsþds q q # q þ q ðt sþ q q qþ ds þ q ¼ q þ q ðq þ Þt qþ þ q q f q q ðsþds f q q ðsþds: ð3:3þ Similarly, for k $ with s ¼ s; we have s # sk s s f ðsþ f ðs k Þ p ffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ds k ds t s s k s k sk sk ðt sþ q qþ ðsk s k Þ q f q q ðsþ f q q ðsk Þds k ds qþ dsk ds q q kgð # q þ Gð q ðq þ Þ qþ Þ kþ þ qþþt qþ Gð þ kþ qþ Þ þ q q qþ q R t f q q ðsþds ðk þ Þ! kþ : Therefore, it follows from (3.) that supku N ðtþ u M ðtþk # supv N;M 6 t 4 þ q þ q þ q X n q k¼ X n k¼ ðq þ Þ ðcðtþþ k T k # f q q ðsþds k! kgð Gð qþ Þ þ qþ ÞT k Gð þ k qþ Þ qþ ðcðtþþ k

18 MARCEL DEKKER, INC. 7 MADISON AVENUE NEW YORK, NY 6 3 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc. 436 Kouritzin, Long, and Sun supku N ðtþ u M ðtþk ðcðtþþ nþ q þ Gð 4 q ðq þ Þ þ ngð qþ Þ nþ þ qþþt qþ Gð þ nþ qþ Þ þ q q T nþ # f q q ðsþds : ð3:4þ ðn þ Þ! Let hðtþ ¼ T f q q ðsþds: By applying Hölder s inequality, Hypothesis (i) and Lemma 3.3, we have T h i E ½hðTÞŠ ¼ f q q ðsþ ds ¼ E T E E T E E # C ð þju N ðs; yþj q þju M ðs; yþj q Þ rðyþdy ð þju N ðs; yþj q þju M ðs; yþj q Þdy ds q q ds # C ðtþ, : ð3:5þ sup ku N ðtþ u M ðtþk is almost surely finite for fixed N and M by the fact that t! ku N ðtþk is càdlàg and the argument on page of Billingsley. [] Thus, it is easy to see that the last term on the right hand side of (3.4) tends to zero almost surely as n!: So, by letting n!in (3.4), we find that supku N ðtþ u M ðtþk # supv N;M Bq þ q ðq þ Þ X þ q k¼ kgð Gð qþ Þ þ qþ ÞT qþ k ðcðtþþ k Gð þ k qþ Þ þ q q exp{cðtþhðtþ} A; ð3:6þ which tends to zero in probability as N; M! by (3.5) and the fact that

19 MARCEL DEKKER, INC. 7 MADISON AVENUE NEW YORK, NY 6 3 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc. Markov Chain Approximations 437 sup V N;M t converges to zero in probability. It follows that for each positive integer n there exists a number N n such that P supku N ðtþ u M ðtþk $!, n n if N; M $ N n : Without loss of generality, we may assume that {N n } is strictly increasing. Let ( ) A n ¼ v [ V : supku N i ðtþ u N j ðtþk, for all i; j $ n n and A ¼ lim sup A n : It is easy to find that P ðaþ¼: Then, for each v [ A; lim supku N i ðt; v Þ u N j ðt; v Þk ¼ : i;j! Note that ðd H ½; TŠ; sup k kþ is a complete non-separable metric space. Thus, for each v [ A; there exists wðv Þ [ D H ½; TŠ such that lim supku N j ðt; v Þ wðt; v Þk ¼ : j! ð3:7þ For v not in A, we define w(v ) as any fixed point of D H [,T ]. By the measurability of w(t, ) for each t [ ½; TŠ and the càdlàg property, we have that w : V! ðd H ½; TŠ; dþ is measurable. Finally, it is easy to see that P!! supku N ðtþ wðtþk. # P supku N ðtþ u N j ðtþk.! þ P supku N j ðtþ wðtþk. : ð3:8þ If we choose N j. N and N sufficiently large, then the right hand side of (3.8) can be made arbitrarily small. This means that sup ku N ðtþ wðtþk! in probability as N!. A

20 MARCEL DEKKER, INC. 7 MADISON AVENUE NEW YORK, NY 6 3 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc. 438 Lemma 3.7. # b # q Kouritzin, Long, and Sun Let w be the limit process from Lemma 3.6. Then for supe kw b ðtþ; l # CðTÞ, : ð3:9þ Proof. By Lemma 3.6, there is a subsequence {N j } such that u N j ðtþ! wðtþ in H almost surely for each t [ ½; TŠ: It follows that ðu N j ðt; xþþ b! w b ðt; xþ a.e. x [ E almost surely. Therefore, by Tonelli s theorem and Fatou s lemma, we find that E kw b ðtþ; l ¼ E w b ðt; xþrðxþdx # lim infe ðu N j ðt; xþþ b rðxþdx E E j! # L L sup N sup ke ðu N ðtþþ b k # CðTÞ, : A Finally, we are in a position to prove Theorem 3.. Proof of Theorem 3.. We have already proved that sup ku N ðtþ wðtþk! in probability in Lemma 3.6. Thus, we only need to show that w(t)is a unique mild solution to (.4). By (.5), we have with wðþ ¼u ; wðtþ ¼TðtÞwðÞþ þ X4 Tðt sþrðwðsþþds þ Tðt sþdqðsþ i NðtÞ; ð3:þ where N ðtþ ¼wðtÞ Tðt sþdqðsþ u N ðtþ and N ðtþ ¼ðT N ðtþu N ðþ TðtÞwðÞÞ þ Y N ðtþ; 3 N ðtþ ¼ T N ðt sþrðu N ðsþþds 4 N ðtþ ¼ Tðt sþ Rðu N ðsþþ RðwðsÞÞ ds: Tðt sþrðu N ðsþþds; T N ðt sþdq N ðsþ ; Whereas sup k N ðtþk! in probability P by Lemmas 3.5 and 3.6,

21 MARCEL DEKKER, INC. 7 MADISON AVENUE NEW YORK, NY 6 3 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc. Markov Chain Approximations 439 sup k 3 NðtÞk! in probability by (3.9). By Trotter-Kato theorem and Lemma 3.4, we have that sup k N ðtþk! in probability P. For 4 NðtÞ; we have by the argument in (3.) and (3.3) that k 4 N ðtþk # ktðt sþ Rðu N ðsþþ RðwðsÞÞ kds # CðTÞsupku N ðsþ wðsþk # CðTÞsupku N ðsþ wðsþk ðt sþ gðsþds ðq þ Þ T q qþ þ q q where : gðsþ ¼ ð þju N ðs; yþj q þjwðs; yþj q Þ rðyþdy E T g q q ðsþds ; R T By Lemmas 3.3 and 3.7, we easily find that E q ðsþds, : Thus, by Lemma 3.6, it follows that sup k 4 N ðtþk! in probability P. Now, from (3.), we find that w(t) is a mild solution of (.4). The last task is to prove the uniqueness. If R is Lipschitz, then the uniqueness follows from standard arguments. For the non-lipschitz case, we define for each n [ N 8 < RðxÞ if jxj # n; R n ðxþ ¼ : otherwise: R nx jxj g q Then, R n is Lipschitz by (.3). Let us denote by u n the solution of (.4) with R n instead of R. Let t n ¼ inf{t : ku n ðtþk # n}: Then, {t n } is a non-decreasing sequence of stopping times and u nþ ðtþ ¼u n ðtþ; ;t # t n : Let t ¼ sup n t n and uðtþ ¼u n ðtþ; ;t # t n : Then, u(t) is a unique solution to (.4) up to time t. We shall prove that t ¼ a:s: For any t. ; we have by (.) some constant C independent of n such that u n ðt; xþ ¼TðtÞuð; xþþ # TðtÞuð; xþþ # kuðþk þ Ct þ C Tðt sþr n ðu n ðs; xþþds þ Tðt sþr þ n ðu n ðs; xþþds þ ku n ðsþk ds þ Xr X N iðtþ j¼ Tðt sþdqðs; xþ Tðt sþdqðs; xþ ku i k A j i :

22 MARCEL DEKKER, INC. 7 MADISON AVENUE NEW YORK, NY 6 3 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc. 44 Kouritzin, Long, and Sun It follows that ku n ðt ^ t n Þk # kuðþk þ Ct ^ t n þ C and consequently, þ Xr N i Xðt^t n Þ j¼ ku i k A j i ; E ku n ðt ^ t n Þk # E kuðþk þ Ct þ C By Gronwall s inequality, we find E ku n ðt ^ t n Þk # CðtÞ, : ^tn þ X r ku ik h i t af i ðdaþ: R þ It is easy to see that P ðt # tþ # P ðt n # tþ # P ku n ðt ^ t n Þk $ n # E ku n ðt ^ t n Þk n # CðtÞ n ku n ðsþk ds E ku n ðs ^ t n Þk ds which tends to zero as n!: So, P ðt # tþ ¼; ;t. ; i.e. t ¼ ; a.s. A ACKNOWLEDGMENTS The authors gratefully acknowledge the support and sponsorship of NSERC, PIms, Lockheed Martin Naval Electronics and Surveillance Systems, Lockheed Martin Canada and Acoustic Positioning Research Inc., through a MITACS center of excellence entitled Prediction in Interacting Systems. REFERENCES. Arnold, L.; Theodosopulu, M. Deterministic limit of the stochastic model of chemical reactions with diffusion. Adv. Appl. Probab. 98,,

23 MARCEL DEKKER, INC. 7 MADISON AVENUE NEW YORK, NY 6 3 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc. Markov Chain Approximations 44. Billingsley, P. Convergence of Probability Measures; Wiley: New York, Blount, D. Comparison of stochastic and deterministic models of a linear chemical reaction with diffusion. Ann. Probab. 99, 9, Blount, D. Density dependent limits for a nonlinear reaction-diffusion model. Ann. Probab. 994,, Blount, D. Diffusion limits for a nonlinear density dependent space-time population model. Ann. Probab. 996, 4, Ethier, S.N.; Kurtz, T.G. Markov Processes, Characterization and Convergence; Wiley: New York, Kallianpur, G.; Xiong, J. Stochastic models of environmental pollution. Adv. Appl. Probab. 994, 6, Kotelenez, P. Law of large numbers and central limit theorem for linear chemical reaction with diffusion. Ann. Probab. 986, 4, Kotelenez, P. High density limit theorems for nonlinear chemical reaction with diffusion. Probab. Th. Rel. Fields 988, 78, 37.. Kouritzin, M.A. Averaging for fundamental solutions of parabolic equations. J. Differ. Eq. 997, 36, Kouritzin, M.A.; Long, H. Convergence of Markov chain approximations to stochastic reaction diffusion equations. Ann. Appl. Probab. 3, in press.. Kurtz, T.G. Limit theorems for sequences of jump Markov processes approximating ordinary differential processes. J. Appl. Probab. 97, 8,

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