A Class of Stochastic Partial Differential Equations for Interacting Superprocesses on a Bounded Domain 1 Introduction

A Class of Stochastic Partial Diﬀerential Equations for Interacting Superprocesses on a Bounded Domain

Yan-Xia Ren, Renming Song and Hao Wang

Abstract

A class of interacting superprocesses on R, called superprocesses with dependent spatial motion (SDSMs), were introduced and studied in Wang [32] and Dawson et al. [9]. In the present paper, we extend this model to allow particles moving in a bounded domain in Rd with killing boundary. We show that under a proper re-scaling, a class of discrete SPDEs for the empirical measure-valued processes generated by branching particle systems subject to the same white noise converge in L2(Ω, F, P) to the SPDE for a SDSM on a bounded domain and the corresponding martingale problem for the SDSMs on a bounded domain is well-posed.

1 Introduction

In this section, we will introduce our model and describe the diﬃculties and challenges we encounter and how we overcome them.

1.1 Model and preliminaries

A class of interacting superprocesses on R known as SDSMs were constructed and studied in Wang [32]. This class of SDSMs includes the super-Brownian motion as a special case. However, SDSMs may exhibit completely diﬀerent features and properties from that of super-Brownian motion. For example, when the underlying dimension is one and the generator is degenerate, the state space of the SDSM consists of purely-atomic measures (see Wang [31]). The study of degenerate SDSMs is closely related to the theory of stochastic ﬂows, see, for example, Dawson et al. [10], [8], Ma et al. [25] and Harris [18].

2000 Mathematics Subject Classiﬁcation: Primary 60J80, 60G57, 60K35; Secondary 60G52

1 For a general reference on stochastic flows, the reader is referred to Kunita [22]. In the present paper, we extend this model to allow particles moving in a bounded domain in Rd with killing boundary. After we establish the existence and uniqueness of the limiting superprocess, we will derive a class of SPDEs for the limiting superprocesses on a bounded domain. Essentially we will follow the basic ideas of Dawson et al. [11] to construct our branching particle systems and to derive the corresponding discrete SPDEs on a bounded domain D in Rd. However, due to the restriction to a bounded domain D our new model raises a sequence of challenges. First of all, since particles are killed upon exiting D, the branching mechanism M n(D × (0, t]) defined by (3.2) is no longer a martingale. Secondly, we have to choose the appropriate form of the infinitesimal generator as (1.2) to take care of the fact that particles are killed upon exiting the domain. Thirdly, in order that (1.2) is an infinitesimal generator of a measure-valued diffusion process, we have to choose a proper domain for it. This forces us to choose the test S ∞ functions φ ∈ D(D) = K⊂D CK (D), the vector space of infinitely differentiable functions ∞ with compact support in D, endowed with the inductive limit of the topologies on CK (D). We will revisit this point in Subsection 1.2. The fourth problem is that Mitoma’s theorem, a basic tool in deriving the limiting SPDE in Dawson et al. [11], is not applicable in the present case, since D(D) is not a Fréchet space. The fifth problem occurs in the proof of the uniqueness of the martingale problem for the SDSM on D when we use Dawson- Kurtz’s duality argument. The difficulty lies in the verification of the invariant property of the dual semigroup. In the following we will explain how to overcome these difficulties. For convenience, the limiting superprocess will be abbreviated as SDSMB. Let D be a bounded domain (i.e., a connected open subset) in Rd. We assume that D is regular, that is, a Brownian motion starting from any boundary point of D will, with probability 1, hit Dc, the complement of D immediately. The dynamic of each Rd-valued particle is described by the following equation: for each k ∈ N,

Z t Z t Z zk(t) − zk(0) = c(zk(s))dBk(s) + h(y − zk(s))W (dy, ds), (1.1) 0 0 Rd

d where {Bk, k ≥ 1} are independent R -valued, standard Brownian motions, W is a Brow- d nian sheet on R (see below for deﬁnition). The processes W and {Bk, k ≥ 1} are assumed T to be independent of each other. h(·) := (hi(·)) = (h1(·), ··· , hd(·)) (written as a column vector, where HT is the transpose of the vector H) is assumed to be an Rd-valued 1 d 2 d Lipschitz function which belongs to L (R ) ∩ L (R ) and c(·) := (cij(·)) (a d × d matrix) is assumed to be an Rd×d-valued Lipschitz function. Then, by the standard Picard’s iteration method we can prove that (1.1) has a unique strong solution which is denoted by

zk(t).

2 Let B(Rd) be the Borel σ-field. By abusing the notation, the Lebesgue measures on d d+1 R and on R will both be denoted by m. Let (Ω, F, {Ft}t≥0, P) be a filtered probability d space with a right continuous filtration {Ft}t≥0. A random set function W on B(R ×R+)

deﬁned on (Ω, F, {Ft}t≥0, P) is called a Brownian sheet or a space-time white noise on Rd if

d (i) for any A ∈ B(R × R+) with m(A) < ∞, W (A) is a Gaussian random variable with mean zero and variance m(A);

d (ii) for any Ai ∈ B(R × R+) with A1 ∩ A2 = ∅ and m(Ai) < ∞, i = 1, 2, W (A1) and

W (A2) are independent and

W (A1 ∪ A2) = W (A1) + W (A2) P-a.s.;

d (iii) for any A ∈ B(R ) with m(A) < ∞, M(A)t := W (A × [0, t]), as a process in t ≥ 0,

is a square-integrable {Ft}-martingale.

For more information on Brownian sheets, the reader is referred to Walsh [30, Chapter 2] and Dawson [6, Section 7.1]. From (1.1), we can see that each particle is subject to a random medium force, which is described by the common Brownian sheet W , and each particle has its own diffusion dynamics, which is described by an individual Brownian motion. In this paper we will use the standard function space notation. L2(Rd) stands for the Hilbert space of square-integrable functions on Rd, and L∞(Rd) the space of bounded measurable functions on Rd. C((Rd)m) and Ck((Rd)m) stand for the space of continuous functions on (Rd)m and the space of continuous functions on (Rd)m with continuous d m k d m derivatives up to and including order k, respectively. Cb((R ) ) and Cb ((R ) ) stand for the space of bounded continuous functions on (Rd)m and the space of bounded continuous functions that have bounded continuous derivatives up to and including order k, respectively. We use Lip(Rd) to denote the space of Lipschitz functions on Rd; that is, f ∈ Lip(Rd) if there is a constant k > 0 such that |f(x) − f(y)| ≤ k|x − y| for every d d d x, y ∈ R . The class of bounded Lipschitz functions on R will be denoted by Lipb(R ). Using the strong solution of (1.1), we can construct a family of branching particle systems in a bounded domain D. For each natural number n ≥ 1, suppose that initially (n) (n) −n there are m0 number of particles located at zi(0), 1 ≤ i ≤ m0 and each has mass θ , where θ > 1 is a fixed constant. These particles evolve according to (1.1) and branch independently at rate γθn with identical offspring distribution in the domain D. Once a particle reaches the boundary of the domain D, it is killed and it disappears from the

3 system. When a particle dies in the domain D, it immediately produces new particles. After branching, the offspring of each particle evolves according to (1.1) and then branches (n) (n) again in D. The common n-th stage branching mechanism q := {qk , k = 0, 1, ···}, (n) where qk stands for the probability that at the n-th stage an individual dies and has k offspring, is assumed to be critical (that is, the average number of offspring is 1), and it can not produce 1 or more than n number of children. Under the assumption that the initial −n Pm(n) d distribution θ k=1 δzk(0) (for any z ∈ R , δz stands for the δ-measure at the point z) of (n) the particles converges weakly to a measure µ0 on D and that the branching function q converges uniformly to a limiting branching function with finite second moment, we will −n P show that, under some additional conditions, the empirical process θ k≥1 δzk(t)1{t<τk} converges weakly to a measure-valued process, where τk = inf{t > 0 : zk(t) ∈/ D} is the first time the k-th particle exiting D.

Let MF (D) denote the Polish space of all ﬁnite measures on D with weak topology and

C(MF (D)) be the space of all continuous functions on MF (D). Based on the assumption that motions are independent of branching, by Itˆo’sformula and a formal calculation we can ﬁnd that the limiting measure-valued processes have the following formal generators (usually called pregenerators. See Section 2 of Dawson [5]):

LF (µ) := AF (µ) + BF (µ), (1.2)

Z 2 1 2 δ F (µ) BF (µ) := 2 γσ 2 µ(dx), (1.3) D δµ(x) and Z µ ¶ Xd ∂2 δF (µ) AF (µ) := 1 (a (x) + ρ (x, x)) µ(dx) (1.4) 2 pq pq ∂x ∂x δµ(x) p,q=1 D p q

Z Z µ ¶ µ ¶ Xd ∂ ∂ δ2F (µ) + 1 ρ (x, y) µ(dx)µ(dy) 2 pq ∂x ∂y δµ(x)δµ(y) p,q=1 D D p q

T T d for F (µ) ∈ D(L) ⊂ C(MF (D)), where for x = (x1, ··· , xd) , y = (y1, ··· , yd) ∈ R ,

Xd apq(x) := cpr(x)cqr(x), (1.5) r=1 Z

ρpq(x, y) := hp(u − x)hq(u − y)du, Rd

4 where the constant γ > 0 above is related to the branching rate of the particle system and σ2 > 0 is the variance of the limiting oﬀspring distribution, the variational derivative is deﬁned by

δF (µ) F (µ + hδ ) − F (µ) := lim x , (1.6) δµ(x) h↓0 h

and D(L), the domain of the pregenerator L, consists of functions of the form

F (µ) = f(hφ1, µi, ··· , hφk, µi)

satisfying following conditions: 2 2 k (1) φi ∈ Cb (D) for 1 ≤ i ≤ k and f ∈ Cb (R ); (2) for any 1 ≤ i ≤ k, φi has compact support in D.

Now let us give the motivation that why we need to choose the domain of the generator (1.2) in this way.

1.2 Motivation for the choice of domain

To simplify the situation and directly show the essential point of the problem, we consider two examples in the ﬁnite dimensional case.

(I) First, we consider a one dimensional reﬂecting Brownian motion. Let {Bt} be a

Ft-Brownian motion on (Ω, F, P). Then Xt = |Bt| is the reﬂecting Brownian motion. The

inﬁnitesimal generator of the semigroup Tt of the reﬂecting Brownian motion is given by (G, D(G)), where 1 00 Gf = f 2 and 2 00 0 D(G) = {f ∈ Cb ([0, ∞)), f is uniformly continuous, f (0) = 0}

2 with Cb ([0, ∞)) being the space of all bounded continuous functions on [0, ∞) with bounded continuous derivatives up to and including order 2. The boundary condition f 0 (0) = 0 is forced upon us by the nature of the reﬂecting Brownian motion. See Section 4.2 of [19] for more information on the reﬂecting Brownian motion . (II) The second example is the coalescing Brownian motion which can be described as

follows. (x1(t), ··· , xm(t)) is called a coalescing Brownian motion if the components move as independent Brownian motions until any pair, say xi(t) and xj(t), (i < j), meet. After that, xj(t) assumes the values of xi(t) and the system continues to evolve in the same

5 fashion. The inﬁnitesimal generator of the transition semigroup of (x1(t), ··· , xm(t)) is given by (Cm, D(Cm)) with

1 X ∂2 Cm = 1 2 {xi=xj } ∂x ∂x 1≤i,j≤m i j

and 2 m 2 m ∂ f D(C ) = {f ∈ Cb (R ): = 0 if xi = xj, for some i 6= j}. ∂xi∂xj

Here again the domain D(Cm) is forced upon us by the nature of the coalescing Brownian motion. For more details on coalescing Brownian motion, one can see [24] and references therein. A similar problem needs to be handled for our measure-valued process. The following 2 k observation may shed some light on the present situation. For any function f ∈ Cb (R ), if we choose φi ∈ D(D), i = 1, ··· , k and

F (µ) = f(hφ1, µi, ··· , hφk, µi),

then, we have δF (µ) = 0 δµ(x) for x ∈ ∂D and δ2F (µ) = 0 δµ(x)δµ(y) S ∞ for either x ∈ ∂D or y ∈ ∂D. So we may choose D(D) = K⊂D CK (D) as the space of the S ∞ test functions for our generator. Recall that the vector space D(D) = K⊂D CK (D) of inﬁnitely diﬀerentiable functions with compact support in D is endowed with the inductive ∞ limit of the topologies on CK (D).

1.3 Basic ideas and organization of the paper

In the usual models (for example, (α, d, β)-superprocesses, see Chapter 4 of Dawson [6] and Perkins [27]), the motions of particles are independent and the motions are independent of branching, thus the particle systems have the following multiplicative property: If

two branching Markov processes evolve independently with initial distribution m1 and m2 respectively, then their sum has the same distribution as the branching process with initial distribution m1 +m2. It is well-known that the log-Laplace functional (or evolution equation) technique can be applied to these models in order to construct the limiting measure-valued process. However, in our model and pregenerator, it is obvious that the

6 motions of particles are not independent, this destroys the multiplicative property. Thus, just as in Wang [32] and Dawson et al [11], the usual log-Laplace functional method is not applicable to our new model. Although Dynkin [13] and Le Gall [23] and other authors have already considered superprocesses on a bounded domain in Rd, in their models particles’ motions are independent and the log-Laplace functional method is applicable. There exists an essential difference between our model and their models. In order to construct the branching particle system in our model, by the Picard’s iteration method we can show that under the assumption that the functions c and h satisfy the Lipschitz condition, the SDE (1.1) has a unique strong solution, which means that (1.1) has a strong solution and the pathwise uniqueness holds. Using the unique strong solutions of (1.1) with different initial positions and that a particle is killed once it exits the domain D, we can construct our branching particle system on D. After proving the tightness of the empirical measure-valued processes constructed from the branching particle system, the existence of the martingale problem for L on D will follow. To prove the uniqueness of the martingale problem for L for the measure-valued interacting process on D, we use a duality method initiated by Dawson and Kurtz [7]. Let n {Pt , t ≥ 0} be the transition semigroup of the underlying motion of n-particles given by (1.1), killed once one of the n particles exits D. Note that the infinitesimal generator of

the n-particles (z1, ··· , zn) is given by

Gnf(x1, ··· , xn) 1 Xn Xd ∂2 := (a (x ) + ρ (x , x )) f(x , ··· , x ) 2 pq i pq i i ∂x ∂x 1 n i=1 p,q=1 ip iq µ ¶ µ ¶ 1 Xn Xd ∂ ∂ + ρpq(xi, xj) f(x1, ··· , xn) 2 ∂xip ∂xjq i6=j,i,j=1 p,q=1 1 Xn Xd ∂2 = Γij (x , ··· , x ) f(x , ··· , x ), (1.7) 2 pq 1 n ∂x ∂x 1 n i,j=1 p,q=1 ip jq

T d where xi = (xi1, ··· , xid) ∈ R and for 1 ≤ i ≤ n,   (apq(xi) + ρpq(xi, xi)) if i = j, Γij (x , ··· , x ) := (1.8) pq 1 n  ρpq(xi, xj) if i 6= j, and © ª 2 d n n f ∈ G(Gn) := f ∈ Cb ((R ) ) : the support of f is a compact subset of D ⊂ D(Gn),

n where D = D × D × · · · × D, the n-fold product, and D(Gn) is the domain of the generator Gn.

7 The remainder of this paper is organized as follows. Section 2 is devoted to the construction of the branching particle system and the derivation of a discrete SPDE for the empirical measure-valued processes. In Section 3, the tightness of the corresponding empirical measure-valued processes and the uniqueness of the SDSMB will be discussed. Then, we prove the L2-convergence of each term in the discrete SPDE and derive a SPDE for the SDSMB. Finally we use Dawson-Kurtz’s duality method to show that the martingale problem for the generator corresponding to the SPDE is well-posed.

2 Branching Particle Systems

In order to construct the branching particle system, we need to introduce an index set to identify each particle in the branching tree structure. Let < be the set of all multi-

indices, i.e., strings of the form ξ = n1 ⊕ n2 ⊕ · · · ⊕ nk, where the ni’s are non-negative integers. Let |ξ| denote the length of ξ. We provide < with the arboreal ordering: m1 ⊕ m2 ⊕ · · · ⊕ mp ≺ n1 ⊕ n2 ⊕ · · · ⊕ nq if and only if p ≤ q and m1 = n1, ··· , mp = np. If |ξ| = p, then ξ has exactly p − 1 predecessors, which we shall denote respectively by ξ − 1, ξ − 2, ··· , ξ − |ξ| + 1. For example, with ξ = 6 ⊕ 18 ⊕ 7 ⊕ 9, we get ξ − 1 = 6 ⊕ 18 ⊕ 7, ξ − 2 = 6 ⊕ 18 and ξ − 3 = 6. We also deﬁne an ⊕ operation on < as follows: if η ∈ < and |η| = m, for any given non-negative integer k, η ⊕ k ∈ < and η ⊕ k is an index for a particle in the (m + 1)-th generation. For example, when η = 3 ⊕ 8 ⊕ 17 ⊕ 2 and k = 1, we have η ⊕ k = 3 ⊕ 8 ⊕ 17 ⊕ 2 ⊕ 1. T d Let {Bξ = (Bξ1, ··· ,Bξd) , ξ ∈ <} be an independent family of standard R -valued

Brownian motions, where Bξk is the k-th component of the d-dimensional Brownian mo- d tion Bξ, and W a Brownian sheet on R . Assume that W and {Bξ, ξ ∈ <} are deﬁned on a common ﬁltered probability space (Ω, F, {Ft}t≥0, P), and independent of each other.

For every index ξ ∈ < and initial data zξ(0), by Picard’s iteration method (see Lemma

3.1 of Dawson et al. [9]), one can easily show that there is a unique strong solution zξ(t) to the equation

Z t Z t Z zξ(t) = zξ(0) + c(zξ(s))dBξ(s) + h(y − zξ(s))W (dy, ds). (2.1) 0 0 Rd

Since the strong solution of (2.1) only depends on the initial state zξ(0), the Brownian motion Bξ := {Bξ(t): t ≥ 0} and the common W , we can write the strong solution of d (2.1) as zξ(t) = Φ(zξ(0),Bξ, t) for some measurable R -valued map Φ (we omit W from the notation as it is selected and ﬁxed once and for all). Let ∂ := ∂ . For each φ ∈ G(G ), p ∂xp 1 we have by Itˆo’sformula that for every t > 0,

8 φ(z (t)) − φ(z (0)) (2.2) ξ " ξ Xd Z t Xd = (∂pφ(zξ(s))) cpi(zξ(s))dBξi(s) p=1 0 i=1 Z t Z ¸ + ∂pφ(zξ(s))hp(y − zξ(s))W (dy, ds) 0 Rd Z Ã ! 1 Xd t Xd + ∂ ∂ φ(z (s)) c (z (s))c (z (s)) ds 2 p q ξ pi ξ qi ξ p,q=1 0 i=1 Z Z 1 Xd t + (∂p∂qφ(zξ(s))) hp(y − zξ(s))hq(y − zξ(s))dyds. 2 d p,q=1 0 R We now consider the branching particle systems in which each particle’s spatial motion is modeled by the SDE (2.1). For every positive integer n ≥ 1, there is an initial system of (n) n n m0 particles. Each particle has mass 1/θ and branches independently at rate γθ . Let (n) qk denote the probability of having k oﬀsprings when a particle dies in D. The sequence (n) {qk } is assumed to satisfy the following conditions:

(n) qk = 0 if k = 1 or k ≥ n + 1, and Xn (n) (n) kqk = 1 and lim sup |qk − pk| = 0, n→∞ k≥0 k=0

where {pk, k = 0, 1, 2, ···} is the limiting oﬀspring distribution which is assumed to satisfy following conditions: X∞ X∞ 2 p1 = 0, kpk = 1 and m2 := k pk < ∞. k=0 k=0

(n) Pn 4 (n) (n) Let mc := k=0(k − 1) qk . The sequence {mc , n ≥ 1} may be unbounded, but we assume that m(n) lim c = 0 for any θ > 1. n→∞ θ2n We will see that the limiting offspring distribution is the offspring distribution of the SDSM on a bounded domain, the limiting measure-valued process that we will construct. (n) n We assume that m0 ≤ ~ θ , where ~ > 0 and θ > 1 are fixed constants. Define (n) Pn 2 (n) 2 n 2 2 2 m2 := k=0 k qk , σn := m2 − 1 and σ := m2 − 1. Note that σn and σ are the variance of the n-th stage and the limiting offspring distribution, respectively. We have 2 2 2 σn < ∞ and limn→∞ σn = σ .

9 For a ﬁxed stage n ≥ 1, let ξ ∈ < and x be the death location of the particle ξ, (n) (n) (n) {Oξ : ξ ∈ <} be a family of i.i.d. random variables with P(Oξ = k) = qk for x ∈ D (n) (n) and k = 0, 1, 2, ··· , otherwise P(Oξ = 0) = 1 for x∈ / D and {Cξ , ξ ∈ <} be a family of i.i.d. real-valued exponential random variables with parameter γθn, which will serve as (n) (n) lifetimes of the particles. We assume W , {Bξ, ξ ∈ <}, {Cξ , ξ ∈ <} and {Oξ , ξ ∈ <} are all independent. In our model, once the particle ξ exits D, it is killed immediately and disappears from the system. In the remainder of this section we are only concerned with stage n. To simplify our notation, we will use the convention of dropping the superscript (n) from the random variables. In later sections we will continue this convention for some random variables such as locations, birth times and death times. This will not cause any confusion, since the stage should be clear from the context. If x, the death location of the particle ξ − 1, belongs to D, then the birth time β(ξ) of the particle ξ is given by   P|ξ|−1 Cξ−j, if Oξ−j ≥ 2 for every j = 1, ··· , |ξ| − 1 ; β(ξ) := j=1  ∞, otherwise.

The death time of the particle ξ is given by ζ(ξ) = β(ξ)+Cξ and the indicator function

of the lifespan of ξ is denoted by `ξ(t) := 1[β(ξ),ζ(ξ))(t).

Recall that ∂ denotes the cemetery point. Deﬁne xξ(t) = ∂ if either t < β(ξ) or t ≥ ζ(ξ). We make the convention that any function f deﬁned on D is automatically extended to D ∪ {∂} by setting f(∂) = 0 — this allows us to keep track of only those particles that are alive at any given time. P (n) (n) n m0 To avoid the trivial case, we assume that µ0 ∈ MF (D). Let µ0 := (1/θ ) ξ=1 δxξ(0) (n) be constructed such that µ0 ⇒ µ0 as n → ∞. We are thus provided with a collection of initial starting points {xξ(0)} for each n ≥ 1.

For a given starting point a ∈ D, let τξ(a) := inf{t : Φ(a, Bξ, t) ∈/ D} be the first exit time of the diffusion process Φ(a, Bξ, t) from the domain D, where Φ is defined in n (n) the paragraph below (2.1). Let N1 := {1, 2, ··· , m0 } be the set of indices for the first n generation of particles. For any ξ ∈ N1 ∩ <, if xξ(0) ∈ D, define ( Φ(xξ(0),Bξ, t), t ∈ [0,Cξ ∧ τξ(xξ(0))), xξ(t) := (2.3) ∂, t ≥ Cξ ∧ τξ(xξ(0)),

and n xξ(t) ≡ ∂ for any ξ ∈ (N \N1 ) ∩ < and t ≥ 0.

10 n If ξ0 ∈ N1 ∩< and xξ0 (ζ(ξ0)−) ∈ ∂D, then xξ(t) ≡ ∂ for any ξ Â ξ0 and any t ≥ ζ(ξ0).

Otherwise, if xξ0 (ζ(ξ0)−) ∈ D and Oξ0 (ω) = k ≥ 2, deﬁne for every ξ ∈ {ξ0 ⊕ i; i = 1, 2, ··· , k}, (

Φ(xξ0 (ζ(ξ0)−),Bξ, t), t ∈ [β(ξ), ζ(ξ) ∧ τξ(xξ0 (ζ(ξ0)−))), xξ(t) := (2.4) ∂, t ≥ ζ(ξ) ∧ τξ(xξ0 (ζ(ξ0)−)).

If Oξ0 (ω) = 0, deﬁne xξ(t) ≡ ∂ for 0 ≤ t < ∞ and ξ ∈ {ξ0 ⊕ i : i ≥ 1}. n More generally for any integer m ≥ 1, let Nm ⊂ < be the set of all indices for the n particles in the m-th generation. If ξ0 ∈ Nm and if xξ0 (ζ(ξ0)−) ∈ ∂D, then xξ(t) ≡ ∂ for

any ξ Â ξ0 and any t ≥ ζ(ξ0). Otherwise, if xξ0 (ζ(ξ0)−) ∈ D and Oξ0 (ω) = k ≥ 2, deﬁne

for ξ ∈ {ξ0 ⊕ i; i = 1, 2, ··· , k} (

Φ(xξ0 (ζ(ξ0)−),Bξ, t), t ∈ [β(ξ), ζ(ξ) ∧ τξ(xξ0 (ζ(ξ0)−))), xξ(t) := (2.5) ∂, t ≥ ζ(ξ) ∧ τξ(xξ0 (ζ(ξ0)−)).

If Oξ0 (ω) = 0, deﬁne

xξ(t) ≡ ∂ for 0 ≤ t < ∞ and for ξ ∈ {ξ0 ⊕ i : i ≥ 1}.

Continuing in this way, we obtainn a branching tree of particleso for any given ω with random initial state taking values in x1(0), x2(0), ··· , x (n) (0) . This gives us our branching m0 particle systems in D ∪ ∂, where particles undergo a ﬁnite-variance branching at independent exponential times and have interacting spatial motions powered by diﬀusions and a common white noise.

3 Tightness, Uniqueness, and SPDE for SDSMB

Recall that {xξ} is the branching particle system constructed in the last section. Deﬁne its associated empirical process by 1 X µ(n)(A) := δ (A) for A ∈ B(D), (3.1) t θn xξ(t) ξ∈<

where B(D) denotes the family of Borel subsets of D. In the following, we will show that (n) {µt , t ≥ 0} converges weakly as n → ∞ and its weak limit is the SDSM on D. For any t > 0 and A ∈ B(D), deﬁne

X [O(n) − 1] M (n)(A × (0, t]) := ξ 1 , (3.2) θn {xξ(ζ(ξ)−)∈A, ζ(ξ)≤t} ξ∈<

11 which describes the space-time related branching in the set A up to time t. Since in the present model the branching particle system and the related superprocess are restricted to a bounded domain in Rd, the framework based on the whole space Rd (for example, Mitoma [26]) is no longer suitable for our new situation. In order to discuss the weak convergence of our empirical measure-valued processes, we introduce some new notation. Let Q be a nonempty open subset of Rd and let C∞(Q) be the set of real-valued functions on Q with continuous derivatives of all orders. For any compact subset K of ∞ ∞ Q, let CK (Q) be the set of functions in C (Q) with support in K. Equipped with the topology given by the seminorms

α pi(φ) := sup{|∂ φ(x)| : x ∈ K, |α| ≤ i}, i ≥ 0,

∞ CK (Q) is a nuclear Fréchet space (see Schaefer [28] and AI-Gwaiz [1] ). Let D(Q) = S ∞ K⊂Q CK (Q) be the vector space of infinitely differentiable functions with compact sup- ∞ port in Q, endowed with the inductive limit of the topologies on CK (Q). Then, D(Q) is usually called the Schwartz space of test functions on Q. Its topological dual, D 0(Q), is the vector space of all distributions or continuous linear functionals on D(Q). D 0(Q) is called the Schwartz space of distributions on Q. ( For more details, the reader is referred to Schwartz [29], Barros-Neto [3] or AI-Gwaiz [1].) Let S(Rd) be the Schwartz space of rapidly decreasing test functions and S0(Rd) be the dual space of S(Rd), the space of Schwartz tempered distributions. Mitoma’s theorem ([26]) provides a convenient tool for studying the weak convergence of measure-valued processes. It is applicable to càdlàgprocesses whose state space is the dual of a nuclear Fréchet space. A typical case is the S0(Rd)-valued processes. However, in our case, D(D) is not a Fréchet space(see AI-Gwaiz [1]). Therefore, Mitoma’s theorem is not applicable to D 0(D)-valued processes. Fortunately Fouque [16] has proved a nice generalization of Mitoma’s theorem to the case which is applicable to càdlàgprocesses whose state space is the dual of an inductive limit topological space of a sequence of nuclear Fréchet spaces. So this works for D 0(D)-valued càdlàgprocesses. Since every Radon measure on D defines a 0 distribution on D, we have MF (D) ⊂ D (D). Note that for a given bounded domain D in Rd, (2.2) implies that for every φ ∈ D(D), 1 hφ, µ(n)i − hφ, µ(n)i = √ U (n)(φ) + X(n)(φ) + Y (n)(φ) + M (n)(φ), (3.3) t 0 θn t t t t where, recall that `ξ(s) = 1[β(ξ), ζ(ξ))(s),

12 X Xd Z t (n) 1 U (φ) := √ `ξ(s) ∂pφ(xξ(s))cpi(xξ(s))dBξi(s) , t n θ ξ∈< p,i=1 0 Xd Z t Z (n) (n) Xt (φ) := hhp(y − ·)∂pφ(·), µs iW (dy, ds) , d p=1 0 R * " # + Xd Z t Xd Z (n) 1 (n) Yt (φ) := 2 ∂p∂qφ(·) cpi(·)cqi(·) + hp(y − ·)hq(y − ·)dy , µs ds, d p,q=1 0 i=1 R Z t Z X (n) (n) (n) [Oξ − 1] Mt (φ) := φ(x)M (dx, ds) = n φ(xξ(ζ(ξ)−)) 1{ζ(ξ)≤t} d θ 0 R ξ∈<

The four terms in (3.3) represent the respective contributions to the overall motion (n) (n) of the finite particle system hφ, µt i in D by the individual Brownian motions (Ut (φ)), (n) the random medium (Xt (φ)), the mean effect of interactive and diffusive dynamics (n) (n) (Yt (φ)), the branching mechanism (Mt (φ)). Using a result of Dynkin ([12] p.325, Theorem 10.25), we immediately get the following theorem.

(n) Theorem 3.1 For any n ∈ N, µt defined by (3.1) is a right continuous strong Markov process which is the unique strong solution of (3.3) in the sense that it is a unique solution (n) (n) of (3.3) for a given probability space (Ω, F, P) and given W , {Bξ},{Cξ }, {Oξ } defined (n) on (Ω, F, P). Furthermore, {µt ; t ≥ 0} are all defined on the common probability space (Ω, F, P).

(n) For each t ≥ 0, let Ft denote the σ-algebra generated by the collection of processes n o (n) (n) (n) (n) (n) µt (φ),Ut (φ),Xt (φ),Yt (φ),Mt (φ), φ ∈ D(D), t ≥ 0 .

(n) (n) (n) Note that according to our assumption, the fourth moment of Oξ , mc := E{(Oξ − (n) 4 mc 1) }, is ﬁnite and limn→∞ θ2n = 0 for any θ > 1.

Lemma 3.2 With the notation above, we have the following.

(n) (n) (i) For every φ ∈ D(D), M (φ) := {Mt (φ), t ≥ 0} is a purely discontinuous square integrable martingale with

Z t (n) 2 2 (n) hM (φ)it = γσn hφ , µu idu for every t ≥ 0. 0

13 (ii) For any t ≥ 0 and n ≥ 1, we have · ¸ (n) 2 (n) 2 2 (n) E sup h1, µs i ≤ 2h1, µ0 i + 8γσnth1, µ0 i. 0≤s≤t Furthermore, there is a constant κ > 0 such that for every t ≥ 0, · ¸ ³ (n) 4 3 6 3 (n) E sup h1, µs i ≤ κ γ σnt h1, µ0 i 0≤s≤t ! γm(n) +γ2σ4t2h1, µ(n)i2 + c th1, µ(n)i + h1, µ(n)i4 . n 0 θ2n 0 0

(n) (iii) {µt ; t ≥ 0} deﬁned by (3.1) is tight as a family of processes with sample paths in D([0, ∞), D 0(D)).

(n) Proof: (i) Recall that, for each n ≥ 1, {Cξ , ξ ∈ <} is a family of i.i.d. exponential n (n) random variables with parameter γθ , {Oξ − 1, ξ ∈ <} is a family of i.i.d random © (n) ª variables with zero mean, and these two families are independent. Thus E Mt (φ) = (n) 0 for every t > 0 and φ ∈ D(D). Since this is valid for any initial distribution µ0 , by (n) the Markov property of {µt , t ≥ 0}, we have for every t, s > 0, h ¯ i h i (n) (n) ¯ (n) (n) (n) E Mt+s(φ) − Mt (φ) F t = E (n) Ms (φ) − M0 (φ) = 0. µt This shows that M (n)(φ) is a martingale. Clearly it is purely discontinuous. · ¸ 2 E φ (xξ(ζ(ξ)−)); ζ(ξ) ≤ t · ¸ (n) 2 (n) = E 1[0,t](β(ξ) + Cξ )φ (xξ((β(ξ) + Cξ )−))

· Z ∞ ¸ 2 n −γθnu = E 1[0,t](β(ξ) + u)φ (xξ((β(ξ) + u)−))γθ e du 0 · Z ∞ ¸ 2 n = E 1[0,t](β(ξ) + u)φ (xξ((β(ξ) + u)−))γθ 1{C(n)>u}du 0 ξ (by independence) · Z ∞ ¸ 2 n = E 1[0,t](β(ξ) + u)φ (xξ((β(ξ) + u)))γθ du 0 (by the deﬁnition of xξ) · Z ∞ ¸ 2 n = E 1[0,t](v)φ (xξ(v))γθ dv β(ξ) · Z t ¸ n 2 = γθ E 1[β(ξ),ζ(ξ))(v)φ (xξ(v))dv . 0

14 (n) (n) (n) As {Cξ , ξ ∈ <} and {Oξ , ξ ∈ <} are all independent and EOξ = 1, we conclude that h i X · ¸ (n) 2 −2n © (n) 2ª 2 E Mt (φ) = θ E (Oξ − 1) E φ (xξ(ζ(ξ)−))1(ζ(ξ)≤t) (3.4) ξ∈< · ¸ X Z t −2n 2 n 2 = θ σn γθ E 1[β(ξ),ζ(ξ))(v)φ (xξ(v))dv ξ∈< 0 · Z t ¸ 2 2 (n) = γσnE hφ , µv i dv . 0

(n) Note that the identity (3.4) holds for any initial distribution µ0 . By the Markov property (n) of {µt , t ≥ 0} again, we have for every t, s > 0, · Z ¸ t+s ¯ (n) 2 (n) 2 2 2 (n) ¯ (n) E Mt+s(φ) − Mt (φ) − γσn hφ , µv i dv F t t · Z s ¸ (n) 2 2 2 (n) = Eµ(n) Ms (φ) − γσn hφ , µv i dv = 0. t 0 R (n) 2 2 t 2 (n) This shows that Mt (φ) − γσn 0 hφ , µv i dv is a martingale. Hence we conclude that M (n)(φ) is a purely discontinuous square integrable martingale with

Z t (n) 2 2 (n) hM (φ)it = γσn hφ , µu idu for every t ≥ 0. 0 (ii) The proof of this part is related the total number of particles of the system. Since the total number of particles of the system with boundary is bounded by the total number of particles of the system without boundary, in the following we only need to prove the result for the system without boundary. As in Section 2, we can reconstruct the branching particle systems without boundary as follows. For each ξ ∈ <, let ( Φ(ˆxξ−1(ζ(ξ − 1)−),Bξ, t), t ∈ [β(ξ), ζ(ξ)), xˆξ(t) := (3.5) ∂, t ≥ ζ(ξ).

Deﬁne 1 X µˆ(n)(A) := δ (A) for A ∈ B(Rd), (3.6) t θn xˆξ(t) ξ∈< For any t > 0 and A ∈ B(Rd), deﬁne

X [O(n) − 1] Mˆ (n)(A × (0, t]) := ξ 1 , (3.7) θn {xˆξ(ζ(ξ)−)∈A, ζ(ξ)≤t} ξ∈<

15 d (n) (n) ˆ (n) Therefore, the particles {xˆξ} live in R . Since h1, µˆt − µˆ0 i = Mt (1) is a zero-mean martingale (for the systems without boundary), by Doob’s maximal inequality, we have · ¸ · ¸ (n) 2 ˆ (n) 2 (n) 2 E sup h1, µˆs i ≤ 2E sup Ms (1) + 2h1, µˆ0 i 0≤s≤t h0≤s≤t i ˆ (n) 2 (n) 2 ≤ 8E Mt (1) + 2h1, µˆ0 i

2 (n) (n) 2 ≤ 8γσnt h1, µˆ0 i + 2h1, µˆ0 i .

P O(n)−1 ˆ (n) ξ (n) Note that Mt (1) = ξ∈< θn 1{ζ(ξ)≤t} is a purely discontinuous martingale and {Oξ − (n) 1, ξ ∈ <} are i.i.d random variables with zero mean and are independent of {Cξ , ξ ∈ <}. Thus Ã ! · ¸ · (n) ¸ ³ ´4 X (O − 1)2 (O(n) − 1)2 E Mˆ (n)(1) = E ξ 1 η 1 t θ2n {ζ(ξ)≤t} θ2n {ζ(η)≤t} ξ,η∈<,ξ6=η · ¸ X (O(n) − 1)4 +E ξ 1 θ4n {ζ(ξ)≤t} ξ∈< · ¸ · ¸ σ4 X m(n) X = n E 1 1 + c E 1 θ4n {ζ(ξ)≤t} {ζ(η)≤t} θ4n {ζ(ξ)≤t} ξ,η∈<,ξ6=η ξ∈< · ¸ · Z ¸ σ4 X γm(n) t = n E 1 1 + c E h1, µˆ(n)idv . θ4n {ζ(ξ)≤t} {ζ(η)≤t} θ2n v ξ,η∈<,ξ6=η 0

(n) (n) For ξ, η ∈ < with ξ 6= η, Cξ and Cη are independent and so £ ¤ E 1 1 ·{ζ(ξ)≤t} {ζ(η)≤t} ¸ (n) (n) = E 1[0,t](β(ξ) + Cξ ) 1[0,t](β(η) + Cη )

· Z ∞ Z ∞ 2 2n = γ θ E 1[0,t](β(ξ) + u) 0 0 ¸ −γθnu −γθnv ·1[0,t](β(η) + v)e e dudv

· Z ∞ Z ∞ 2 2n ≤ γ θ E 1[0,t](β(ξ) + u) 0 0 ¸

·1[0,t](β(η) + v)1{xξ(β(ξ)+u)6=∂}1{xη(β(η)+u)6=∂}dudv

· µZ ∞ ¶ µZ ∞ ¶ ¸ 2 2n = γ θ E 1[0,t](r)1{xξ(r)6=∂}dr 1[0,t](s)1{xη(s)6=∂}ds β(ξ) β(η) · µZ t ¶ µZ t ¶ ¸ 2 2n = γ θ E 1[β(ξ), ζ(ξ))(r)dr 1[β(η), ζ(η))(s)ds . 0 0

16 Therefore X · ¸ E 1{ζ(ξ)≤t}1{ζ(η)≤t} ξ,η∈<,ξ6=η · µ ¶ µ ¶ ¸ X Z t Z t 2 2n ≤ γ θ E 1[β(ξ), ζ(ξ))(r)dr 1[β(η), ζ(η))(s)ds ξ,η∈< 0 0 · Ã !2 ¸ X Z t 2 2n = γ θ E 1[β(ξ), ζ(ξ))(r)dr ξ∈< 0 · µZ t ¶2 ¸ 2 4n (n) = γ θ E h1, µˆr idr . 0 It follows then · µZ ¶2 ¸ · Z ¸ £ ³ ´4 ¤ 4 t (n) t ˆ (n) σn 2 4n (n) γmc (n) E Mt (1) ≤ 4n γ θ E h1, µˆr idr + 2n E h1, µˆv idv θ 0 θ 0 · ¸ (n) 2 4 2 (n) 2 γmc (n) ≤ γ σnt E sup h1, µˆr i + 2n h1, µˆ0 it r∈[0,t] θ ³ ´ γm(n) ≤ γ2σ4t2 8γσ2t h1, µˆ(n)i + 2h1, µˆ(n)i2 + c h1, µˆ(n)it n n 0 0 θ2n 0 γm(n) = 8γ3σ6t3h1, µˆ(n)i + 2γ2σ4t2h1, µˆ(n)i2 + c th1, µˆ(n)i. (3.8) n 0 n 0 θ2n 0 By Doob’s maximal inequality, · ¸ (n) 4 E sup h1, µˆs i 0≤·s≤t ¸ ³ ´4 (n) (n) (n) ≤ E sup h1, µˆs − µˆ0 i + h1, µˆ0 i ·0≤s≤t ¸ ˆ (n) 4 (n) 4 ≤ 8E sup |Ms (1)| + 8h1, µˆ0 i 0≤s≤t· ¸ ³ ´4 4 ˆ (n) (n) 4 ≤ 8(4/3) E Mt (1) + 8h1, µˆ0 i Ã ! γm(n) ≤ κ γ3σ6t3h1, µˆ(n)i + γ2σ4t2h1, µˆ(n)i2 + c th1, µˆ(n)i + h1, µˆ(n)i4 . n 0 n 0 θ2n 0 0

(n) (n) We know thatµ ˆ0 = µ0 ∈ MF (D). Therefore, the conclusion follows. (iii) By Fouque’s Theorem (Fouque [16]), Theorem 4.5.4 in Dawson [5], and part (ii) above, which implies non-explosion in ﬁnite time, we only need to prove that, if we are given ε > 0, T > 0, φ ∈ D(D), and a sequence of stopping times τn bounded by T , then (n) (n) ∀ η > 0, ∃ δ, n0 such that supn≥n0 supt∈[0,δ] P{|µτn+t(φ) − µτn (φ)| > ε} ≤ η.

17 We have by (3.3),

P(|µ(n) (φ) − µ(n)(φ)| > ε) τn+t· τn ¸ 1 ³ ´2 ≤ E µ(n) (φ) − µ(n)(φ) ε2 τn+t τn · 4 1 ³ ´2 ³ ´2 ≤ E U (n) (φ) − U (n)(φ) + X(n) (φ) − X(n)(φ) ε2 θn τn+t τn τn+t τn ¸ ³ ´2 ³ ´2 + Y (n) (φ) − Y (n)(φ) + M (n) (φ) − M (n)(φ) . τ(n)+t τn τn+t τn

Note that by the independence of {Bξ, ξ ∈ <}, · ¸ ³ ´2 (n) (n) E Uτn+t(φ) − Uτn (φ)

d · Z ¸ 1 X X τn+t = E ` (s)[∂ φ(x (s))c (x (s))]2ds θn ξ p ξ pi ξ ξ∈< p,i=1 τn d · Z ¸ X τn+t 2 (n)® = E (∂pφcpi) , µs ds p,i=1 τn Xd · ¸ 2 (n)® ≤ t E sup (∂pφcpi) , µs s≤T +t p,i=1 Xd · ¸ 2 (n)® ≤ t k∂pφcpik∞ E sup 1, µs , s≤T +t p,i=1 while · ¸ ³ ´2 (n) (n) E Xτn+t(φ) − Xτn (φ)

d · Z Z ¸ X τn+t (n) (n) = E hhp(y − ·)∂pφ(·), µs ihhq(y − ·)∂qφ(·), µs idyds d p,q=1 τn R d · Z · Z Z ¸ ¸ X τn+t (n) (n) = E ρpq(w, z)∂pφ(w)∂qφ(z)µs (dw)µs (dz) ds d d p,q=1 τn R R d · Z ¸ X τn+t (n) 2 ≤ {kρpqk∞k∂pφk∞k∂qφk∞}E h1, µs i ds p,q=1 τn Xd · ¸ (n) 2 ≤ t {kρpqk∞k∂pφk∞k∂qφk∞} E sup h1, µs i . s≤T +t p,q=1

18 and · ¸ · d ¸ · Z ¸ ³ ´2 X ¡ ¢ 2 τn+t E Y (n) (φ) − Y (n)(φ) ≤ 1 k(a + ρ )∂ ∂ φk E h1, µ(n)i2ds τn+t τn 2 pq pq p q ∞ s p,q=1 τn · ¸ · ¸ Xd ¡ ¢ 2 1 (n) 2 ≤ t 2 k(apq + ρpq)∂p∂qφk∞ E sup h1, µs i . s≤T +t p,q=1 Finally we have by part (i) of this lemma that · ¸ · Z ¸ ³ ´2 τn+t (n) (n) 2 2 (n) E Mτn+t(φ) − Mτn (φ) = γσnE hφ , µs ids τn · ¸ 2 2 (n) ≤ γσnkφk∞ t E sup h1, µs i . s≤T +t Therefore by part (ii) of this lemma and Lemma 3.4 of Wang [32], we conclude that for every ε > 0, there is a constant c > 0 such that ³ ´ (n) (n) sup sup P |µτn+t(φ) − µτn (φ)| > ε ≤ cδ for every δ > 0, n≥1 t∈[0,δ] which proves (iii). This completes the proof of the lemma. ¤

Theorem 3.3 With the notation above, we have following conclusions: (i) (µ(n),U (n),Y (n),M (n)) is tight on D([0, ∞), (D 0(D))4).

(ii) (A Skorohod representation): Suppose that the joint distribution of

(µ(nm),U (nm),Y (nm),M (nm),W )

converges weakly to the joint distribution of

(µ(0),U (0),Y (0),M (0),W ).

Then, there exist a probability space (Ωe, Fe, Pe) and D([0, ∞), D 0(D))-valued sequences {µe(nm)}, {Ue(nm)}, {Ye (nm)}, {Mf(nm)} and a D([0, ∞), S0(Rd))-valued sequence {Wf(nm)} deﬁned on it, such that

P ◦ (µ(nm),U (nm),Y (nm),M (nm),W )−1 = Pe ◦ (µe(nm), Ue(nm), Ye (nm), Mf(nm), Wf(nm))−1

holds and, Pe-almost surely on D([0, ∞), (D 0(D))4 × S0(Rd)), ³ ´ ³ ´ µe(nm), Ue(nm), Ye (nm), Mf(nm), Wf(nm) → µe(0), Ue(0), Ye (0), Mf(0), Wf(0)

as m → ∞.

19 (iii) There exists a dense subset Ξ ⊂ [0, ∞) such that [0, ∞) \ Ξ is at most countable and for each t ∈ Ξ and each φ ∈ D(D) and each ϕ ∈ S(Rd), as R5-valued processes

(nm) e(nm) e (nm) f(nm) f(nm) (µet (φ), Ut (φ), Yt (φ), Mt (φ), Wt (ϕ)) (0) e(0) e (0) f(0) f(0) → (µet (φ), Ut (φ), Yt (φ), Mt (φ), Wt (ϕ))

2 e e e e(0) in L (Ω, F, P) as m → ∞. Furthermore, let Ft be the σ-algebra generated by (0) (0) (0) (0) (0) d µes (φ), Ues (φ), Yes (φ), Mfs (φ), Wfs (ϕ) for all φ ∈ D(D), all ϕ ∈ S(R ) and f(0) e(0) s ≤ t. Mt (φ) is a continuous, square-integrable Ft -martingale with quadratic variation process

Z t f(0) 2 2 (0) hMt (φ)i = γσ hφ , µeu idu 0

(iv) Wf(0)(dy, ds) and Wf(nm)(dy, ds) are Brownian sheets and for any φ ∈ D(D) the continuous square integrable martingale Z Z Xd t ® e(nm) (nm) f(nm) Xt (φ) := hp(y − ·)∂pφ(·), µes W (dy, ds) d p=1 0 R converges to Z Z Xd t ® e(0) (0) f(0) Xt (φ) := hp(y − ·)∂pφ(·), µes W (dy, ds) d p=1 0 R

in L2(Ωe, Fe, Pe) .

(0) (0) (0) (v) µe = {µet , t ≥ 0} is a solution to the (L, δµ0 )-martingale problem and µe is a continuous process and for any φ ∈ D(D) we have

Z t Z (0) (0) e(0) f(0) µet (φ) − µe0 (φ) = Xt (φ) + φ(x)M (dx, ds) * 0 Rd + Z t Xd 1 (0) + 2 (apq(·) + ρpq(·, ·))∂p∂qφ(·), µes ds . (3.9) 0 p,q=1

Proof: (i) By a theorem of Fouque [16], we only need to prove that, for any φ ∈ D(D), the sequence of laws of

(µ(n)(φ),U (n)(φ),Y (n)(φ),M (n)(φ))

20 is tight in D([0, ∞), R4). This is equivalent to proving that each component and the sum of each pair of components are individually tight in D([0, ∞), R). Since the same idea works for each sequence, we only give the proof for {M (n)(φ)}. By Lemma 3.2 we have ³ ´ 2 Z t 2 2 (n) γσn 2 (n) γσnkφk∞t (n) P Mt (φ) > k ≤ 2 E hφ , µu idu ≤ 2 h1, µ0 i , k 0 k which yields the compact containment condition. Now we use Kurtz’s tightness criterion (cf. Ethier-Kurtz [15] p. 137, Theorem 8.6) to prove the tightness of {M (n)(φ)}. T 2 2 (n) Let γn (δ) := δγσnkφk∞ sup0≤u≤T h1, µu i, then for any 0 ≤ t + δ ≤ T , · ¯ ¸ · Z ¯ ¸ · ¯ ¸ ¯ t+δ ¯ ¯ (n) (n) 2¯ (n) 2 2 (n) ¯ (n) T ¯ (n) E |Mt+δ(φ) − Mt (φ)| ¯Ft = E γσn hφ , µu idu¯Ft ≤ E γn (δ)¯Ft . t T (n) By Lemma 3.2, limδ→0 supn E[γn (δ)] = 0 holds, so {M (φ), n ≥ 1} is tight. d 0 0 0 0 d (ii) Let Ec = D(D) or Ec = S(R ), then Ec = D (D) or Ec = S (R ) respectively. 0 Since Ec is separable and Ec is a completely regular topological space (a nuclear space is separable, cf. Gel’fand-Vilenkin [17]), we can choose a countable dense subset {gi}i∈N of Ec and any enumeration {tj}j∈N of all the rational numbers, then use Theorem 1.7 of

Jakubowski [20] to get that the countable family {fij : i, j ∈ N} of continuous functions 0 0 separate points in D([0, ∞),Ec) (with respect to Skorohod topology on D([0, ∞),Ec)), where 0 fij : x ∈ D([0, ∞),Ec) → fij(x) := arctanhgi, x(tj)i ∈ [−π, π]. 0 0 4 0 d This proves that the space D([0, ∞),Ec), and thus the space D([0, ∞), (D (D)) ×S (R )) satisfy the basic assumption for a version of the Skorohod Representation Theorem due to Jakubowski [21]. d (iii) For each t ∈ Ξ and eachn φ ∈ D(D) and each ϕ ∈ S(R ), from Lemma 3.2 we obtaino (nm) 2 e(nm) 2 e (nm) 2 f(nm) 2 f(nm) 2 the uniform integrability of µet (φ) , Ut (φ) , Yt (φ) , Mt (φ) , Wt (ϕ)) . So (ii) implies their convergence in L2(Ωe, Fe, Pe) as m → ∞. For each φ ∈ D(D), ϕ ∈ S(Rd) 5 and Gi ∈ Cb(R ), and any 0 < t1 ≤ t2, ≤ · · · ≤ tn = s < t with ti, t ∈ Ξ, i = 1, ··· , n, let Yn (nm) (nm) e(nm) e (nm) f(nm) f(nm) f (t1, ··· , tn) := Gi(µeti (φ), Uti (φ), Yti (φ), Mti (φ), Wti (ϕ))). i=1 Then, we have h i e f(nm) f(nm) (nm) E (Mt (φ) − Ms (φ))f (t1, ··· , tn) = 0 (3.10) and ·µ Z t e f(nm) 2 2 2 (nm) f(nm) 2 E Mt (φ) − γσnm hφ , µeu idu − Ms (φ) 0 Z s ¶ ¸ 2 2 (nm) (nm) +γσnm hφ , µeu idu f (t1, ··· , tn) = 0. (3.11) 0

21 2 e e e f(0) By the convergence in L (Ω, F, P) above, this implies that Mt and

Z t f(0) 2 2 2 (0) Mt (φ) − γσ hφ , µeu idu 0

e(0) 2 are Ft -martingales. Let K = supx∈D φ (x). Using (3.8) we can get h i e f(n) f(n) 4 E (Mt (φ) − Ms (φ)) (3.12) h i (n) (n) 4 = E (Mt (φ) − Ms (φ)) " Ã !# (n) X (O − 1)2 (O(n) − 1)2 = E ξ φ2(x (ζ(ξ)−))1 η φ2(x (ζ(η)−))1 θ2n ξ {s<ζ(ξ)≤t} θ2n η {s<ζ(η)≤t} ξ,η∈<,ξ6=η " Ã !# X (O(n) − 1)4 +E ξ φ4(x (ζ(ξ)−))1 θ4n ξ {s<ζ(ξ)≤t} ξ∈< " Ã !# (n) X (O − 1)2 (O(n) − 1)2 ≤ K2E ξ 1 η 1 θ2n {s<ζ(ξ)≤t} θ2n {s<ζ(η)≤t} ξ,η∈<,ξ6=η " Ã !# X (O(n) − 1)4 +K2E ξ 1 θ4n {s<ζ(ξ)≤t} ξ∈< Ã ! γm(n) ≤ K2 8γ3σ6(t − s)3h1, µ(n)i + 2γ2σ4(t − s)2h1, µ(n)i2 + c (t − s)h1, µ(n)i . n 0 n 0 θ2n 0

In particular, for any m ≥ 1 we have h i e f(nm) f(nm) 4 E (Mt (φ) − Ms (φ)) (3.13) Ã ! (nm) 2 3 6 3 (nm) 2 4 2 (nm) 2 γmc (nm) ≤ K 8γ σn (t − s) h1, µ0 i + 2γ σn (t − s) h1, µ0 i + (t − s)h1, µ0 i . m m θ2nm

Let m → ∞ we get h i ¡ ¢ e f(0) f(0) 4 2 3 6 3 2 4 2 2 E (Mt (φ) − Ms (φ)) ≤ K 8γ σ (t − s) h1, µ0i + 2γ σ (t − s) h1, µ0i . (3.14)

f(0) Thus, Mt has a continuous modiﬁcation according to Kolmogorov’s continuity criterion and Z t f(0) 2 2 (0) hMt (φ)i = γσ hφ , µeu idu. 0

(iv) Since W , Wf(0) and Wf(nm) have the same distribution, Wf(0) and Wf(nm) are Brownian sheets. The conclusion follows from (ii) and Theorem 2.1 of Cho [4].

22 e(nm) 2 e 2 e (v) Since Ut (φ) is uniformly integrable, we have P-a.s. and in L (P) 1 e(nm) lim √ Ut (φ) = 0. m→∞ θnm

By taking n → ∞ along the subsequence {nm, m ≥ 1} in (3.3), we have

(0) (0) e(0) e (0) f(0) µet (φ) − µe0 (φ) = Xt (φ) + Yt (φ) + Mt (φ) for every φ ∈ D(D) and t ≥ 0.

As * + Z t Xd e (n) 1 (n) Yt (φ) = 2 (apq(·) + ρpq(·, ·))∂p∂qφ(·), µes ds 0 p,q=1 R R f(n) t f(n) and Mt (φ) = 0 D φ(x)M (dx, ds), we see from (ii) above that * + Z t Xd e (0) 1 (0) Yt (φ) = 2 (apq(·) + ρpq(·, ·))∂p∂qφ(·), µes ds 0 p,q=1 R R f(0) t f(0) (0) and Mt (φ) = 0 D φ(x)M (dx, ds). So, µet satisﬁes (3.9). (0) By Itˆo’sformula, we see that {µet ; t ≥ 0} is a solution to the martingale problem for (0) (L, δµ0 ). The continuity of µe follows from the result of Bakry-Emery [2]. ¤

(0) (0) We see from the theorem above that µe = {µet ; t ≥ 0} is a solution to the martingale problem for (L, δµ0 ). For uniqueness, we will use a duality argument due to Dawson-Kurtz [7]. Before we start the discussion of uniqueness of the martingale problem for L, in the following we will rewrite L into an equivalent form. Recall

LF (µ) := AF (µ) + BF (µ), (3.15)

Z 2 1 2 δ F (µ) BF (µ) := 2 γσ 2 µ(dx), (3.16) D δµ(x) and Z µ ¶ Xd ∂2 δF (µ) AF (µ) := 1 (a (x) + ρ (x, x)) µ(dx) (3.17) 2 pq pq ∂x ∂x δµ(x) p,q=1 D p q

Z Z µ ¶ µ ¶ Xd ∂ ∂ δ2F (µ) + 1 ρ (x, y) µ(dx)µ(dy). 2 pq ∂x ∂y δµ(x)δµ(y) p,q=1 D D p q

23 2 m 2 m Let C0 (D ) be the collection of functions in C (D ) vanishing on the boundary and m 2 m m c d m m outside of D . Therefore, for ∀f ∈ C0 (D ) and ∀x ∈ (D ) := (R ) \ D we have ∞ 2 m 2 m f(x) = 0. For f ∈ ∪m=1C0 (D ), we deﬁne N(f) to be m if f ∈ C0 (D ) and deﬁne Z Z d Fµ(f) := Ff (µ) := ··· f(x1, ··· , xm)µ(dx1) ··· µ(dxm) for µ ∈ MF (R ). Rd Rd

d Such a function Ff is called a monomial function on the space MF (R ). Note that for

such a monomial function Ff ,

NX(f) Z NY(f) ∂Ff (µ) = f(x1, ··· , xj−1, x, xj+1, ··· , xN(f)) µ(dxl), ∂µ(x) d N(f)−1 j=1 (R ) l=1,l6=j 2 ∂ Ff (µ) ∂µ(y)∂µ(x) NX(f) Z NY(f) = f(x1, ··, xj−1, x, xj+1, ··· , xk−1, y, xk+1, ··, xN(f)) µ(dxl). d N(f)−2 j,k=1, j6=k (R ) l=1,l6=j,k

2 m d m For f ∈ C0 (D ), x = (x1, ··· , xm) ∈ (R ) , we deﬁne

1 Xm Xd ∂2 G f(x) := Γij (x , ··· , x ) f(x , ··· , x ) (3.18) m 2 pq 1 m ∂x ∂x 1 m i,j=1 p,q=1 ip jq

T d ij where xi = (xi1, ··· , xid) ∈ R for 1 ≤ i ≤ m and Γpq is deﬁned by (1.8). Then by (1.2), d we have for any monomial function Ff on MF (R ) with N(f) = m,

LFf (µ) = AFf (µ) + BFf (µ) γσ2 Xm = F (µ) + F (µ) Gmf 2 Φjkf j,k=1, j6=k γσ2 Xm ¡ ¢ γσ2 = F (µ) + F (µ) − F (µ) + m(m − 1)F (µ) Gmf 2 Φjkf f 2 f j,k=1, j6=k γσ2 Xm γσ2 = F (G f) + (F (Φ f) − F (f)) + m(m − 1)F (f) µ m 2 µ jk µ 2 µ j,k=1, j6=k 1 := L∗F (f) + γσ2m(m − 1)F (f). µ 2 µ

d m−1 d m−1 Here for j < k,Φjkf is a function on (R ) deﬁned by, for y = (y1, ··· , ym−1) ∈ (R ) ,

Φjkf(y) := Φkj(y) := f(y1, ··· , yj, ··· , yk−1, yj, yk, ··· , ym−1).

24 We know that (1.1) has a unique strong solution. For any positive integer m, let {zi(t), t ≥ 0, 1 ≤ i ≤ m} be a sequence of strong solutions to (1.1) for the given standard Rd-valued d Brownian motions {Bi(t), t ≥ 0, 1 ≤ i ≤ m} and W , a Brownian sheet on R , defined on m (Ω, F, {Ft}t≥0, P). Let Zm(t) := (z1(t), ··· , zm(t)) and let {Pt : t ≥ 0} be the transition m semigroup of Zm(t) killed upon leaving D . Since the coefficients of (1.1) satisfy the m 2 m 2 m Lipschitz condition, Pt is a Feller semigroup and maps C0 (D ) to C0 (D ). m For any given integer m ≥ 1, the law of Zm with initial point x ∈ D will be denoted m by Px, and expectation with respect to Px will be denoted by Ex. Let τ be the first exit m time of Zm from D .

d d Theorem 3.4 Assume that D is a bounded regular domain in R , c ∈ Lipb(R ), h ∈ 2 d 1 d d L (R ) ∩ L (R ) ∩ Lipb(R ) and the diﬀusion matrix (apq)1≤p,q≤d deﬁned by (1.5) is uni- d m formly elliptic, bounded on R . Let {Pt : t ≥ 0} be the transition semigroup for Zm(t) killed upon leaving Dm, that is

m m m Pt f(x) := Ex [f(Zm(t)); t < τ ] for t ≥ 0 and f ∈ Bb(D ).

2 m m 2 m Then for every f ∈ C0 (D ) and t > 0, Pt f(x) as a function of x belongs to C0 (D ). 2 m m Therefore, C0 (D ) is invariant under Pt for every t > 0 and m ≥ 1.

Proof: To apply Theorem 1.6 in part II of [14], we only need to check the uniform ij T d ellipticity of (Γpq). Let ξi = (ξi1, ··· , ξid) denote an arbitrary column vector in R and ij Γ := (Γpq(x1, ··· , xm))1≤i,j≤m,1≤p,q≤d. Since   ξ  1  Xm Xd T T  .  ij (ξ , ··· , ξ )Γ  .  = ξipΓ (x1, ··· , xm)ξjq 1 m   pq i,j=1 p,q=1 ξm Xm Xd Xm Xd = [ξip(apq(xi) + ρpq(xi, xi))ξiq] + ξipρpq(xi, xj)ξjq i=1 p,q=1 i,j=1,i6=j p,q=1 m · d d Z d ¸ X X ¡ X ¢2 ¡ X ¢2 = ξipcpr(xi) + ξiphp(u − xi) du i=1 r=1 p=1 E p=1 Z Xm ¡ Xd ¢¡ Xd ¢ + ξiphp(u − xi) ξjqhq(u − xj) du i,j=1,i6=j E p=1 p=1 m d d Z · m d ¸2 X X ¡ X ¢2 X ¡ X ¢ = ξipcpr(xi) + ξiphp(u − xi) du ≥ 0 (3.19) i=1 r=1 p=1 E i=1 p=1

25 and by the uniform ellipticity assumption of (apq)1≤p,q≤d there exists a positive real number ² > 0 such that for each 1 ≤ i ≤ m

Xd Xd Xd ¡ ¢2 2 ξipcpr(xi) = [ξipapq(xi)ξiq] ≥ ²|ξi| , (3.20) r=1 p=1 p,q=1 p 2 2 where |ξi| = ξi1 + ··· + ξid, the uniform ellipticity of Γ follows. The assumption of 1 d d d h ∈ L (R ) implies that ρ(xi, xj) ∈ Lip(R × R ) for i, j = 1, 2, ··· , m, and therefore i,j d m m Γp,q ∈ Lip((R ) ). By Theorem 1.6 in part II of [14], we have for every f ∈ C0(D ), 2 m there is a u ∈ Cb (D ) such that ∂u − G u = 0 in (0, t) × Dm, (3.21) ∂s m and

u(0, x) = f(x), u(s, x)|(0,t)×∂Dm = 0, (3.22)

m where Gm is the differential operator given by (3.18). Let Dn , n ≥ 1 be a sequence of m m m m m bounded smooth domains such that Dn ⊂ D and Dn ↑ D as n ↑ ∞, and let τn be m the first exit time from Dn . Applying Itô’sformula to s 7→ u(t − s, Zm(s)) (see, e.g. the m calculation for (2.2)), we see that Ls := u(t − s, Zm(s)) is a local martingale on [0, t ∧ τ ),

m i.e., for any fixed m ≥ 1 and any fixed n ≥ 1, (Ls∧t∧τn , s ≥ 0) is a martingale. Since Ls m m m is bounded and τ < ∞ a.s. Px for any x ∈ D , Ls converges to a limit as s → t ∧ τ . Since u is continuous on [0, t] × Dm and satisfies the boundary condition (3.22), the limit must be f(Zm(t))I(t<τ m). Thus

m m u(t, x) = Ex [f(Zm(t)); t < τ ] = Pt f(x).

This proves the theorem. ¤

∞ 2 k Let S = D∪(∪k=1C0 (D )) (disjoint union). We see from the proof of Theorem 3.4 that 2 m Gm coincides on C0 (D ) with the inﬁnitesimal generator of the strong Markov process m ∗ Zm for the motion of m particles given by (2.1) killed upon exiting D . Thus L has the structure of the inﬁnitesimal generator of an S-valued strong Markov process Y , whose dynamics contains the following two mechanisms:

(a) Jumping mechanism: Let {Jt : t ≥ 0} be a nonnegative integer-valued c`adl`agMarkov

process with J0 = m and transition intensities {qi,j} such that

γσ2 qi,i−1 = −qi,i = 2 i(i − 1) and qi,j = 0 for all other pairs (i, j).

26 Thus, {Jt, t ≥ 0} is just the well-known Kingman’s coalescent process. Let τ0 = 0,

τJ0+1 = ∞ and {τk : 1 ≤ k ≤ J0} be the sequence of jump times of {Jt : t ≥ 0}.

Let {Sk : 1 ≤ k ≤ M0} be a sequence of random operators which are conditionally

independent given {Jt : t ≥ 0} and satisfy 1 P{S = Φ |J(τ −) = l} = , 1 ≤ i 6= j ≤ l. k i,j k l(l − 1)

(b) Spatial jump-diﬀusion semigroup: Let B denote the topological union of {L∞((D)m): m = 1, 2, ···} endowed with pointwise convergence on each L∞((D)m). Then

Jτ Jτk k−1 Jτ1 J0 Yt = Pt−τk SkPτk−τk−1 Sk−1 ··· Pτ2−τ1 S1Pτ1 Y0, τk ≤ t < τk+1, 0 ≤ k ≤ J0,

2 m deﬁnes a Markov process Y := {Yt : t ≥ 0} with Y0 ∈ C0 (D ). By Theorem 3.4, process Y takes values in S ⊂ B. Clearly, {(Jt,Yt): t ≥ 0} is also a Markov process.

The duality relationship can be described as follows. Let D(L) be the set of all m 2 m functions of the form Fm,f (µ) = hf, µ i with f ∈ C0 (D ). If {Xt : t ≥ 0} is a solution to the (L, D(L))-martingale problem with X0 = µ0 on a probability space (Ω, F, P), then, by Feynman-Kac formula (see [7]), we have

· µ 2 Z t ¶¸ m Jt γσ E [hf, Xt i] = Em,f hYt, µ0 i exp Js(Js − 1)ds (3.23) 2 0 2 m for any t ≥ 0, f ∈ C0 (D ) and integer m ≥ 1, where the right hand side is the expectation taken on the probability space where the dual process is deﬁned with giving J0 = m and 2 m Y0 = f ∈ C0 (D ). From this, we see that the marginal distribution of X is uniquely determined and hence the law of X is unique (see, e.g. [15, Theorem 4.4.2]). This proves the uniqueness of the martingale problem for L. We summarize these results in the following theorem.

d d Theorem 3.5 Assume that D is a bounded regular domain in R , c ∈ Lipb(R ) and 2 d d h ∈ L (R ) ∩ Lip(R ), and the diﬀusion matrix (apq)1≤p,q≤d deﬁned by (1.5) is uniformly elliptic and bounded on Rd. For the uniqueness for the martingale problem below, assume further that c is bounded below by a strictly positive constant and h ∈ L1(Rd). For any

measure µ ∈ MF (D) with compact support, the (L, δµ)-martingale problem has a unique

solution µt, which is a diﬀusion process and satisﬁes Z t Z µt(φ) − µ0(φ) = Xt(φ) + φ(x)M(dx, ds) * 0 Rd + Z t Xd 1 + 2 (apq(·) + ρpq(·, ·))∂p∂qφ(·), µs ds (3.24) 0 p,q=1

27 for every t > 0 and φ ∈ D(D), where W is a Brownian sheet,

Xd Z t Z Xt(φ) := hhp(y − ·)∂pφ(·), µsi W (dy, ds) d p=1 0 R and M is a square-integrable martingale measure with

Z t 2 2 hM(φ)it = γσ hφ , µuidu for every t > 0 and φ ∈ D(D). 0 Here Z t Z Mt(φ) := φ(y)M(ds, dy) 0 Rd is a square-integrable, continuous {Ft}-martingale, where Ft := σ{µs(f),Ms(f),Xs(f), f ∈ 1 C (D), s ≤ t}. Moreover Xt(φ), Mt(φ) are orthogonal square-integrable, {Ft}-martingales for every φ ∈ D(D).

Acknowledgements. The research of Yan-Xia Ren is supported in part by NNSF of China (Grant No. 10471003). The authors thank the referee for the careful reading of the ﬁrst version of this paper and for useful suggestions for improving the quality of the paper.

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30 Yan-Xia Ren: LMAM School of Mathematical Sciences, Peking University, Beijing, 100871, P. R. China, E-mail: [email protected]

Renming Song: Department of Mathematics, The University of Illinois, Urbana, IL 61801 U.S.A., E-mail: [email protected]

Hao Wang: Department of Mathematics, The University of Oregon, Eugene OR 97403- 1222, U.S.A., E-mail: [email protected]