SUPERPROCESSES WITH DEPENDENT SPATIAL MOTION AND GENERAL BRANCHING DENSITIES

We construct a class of superprocesses by taking the high density limit of a sequence of interacting-branching particle systems. The spatial motion of the superprocess is determined by a system of interacting di(cid:11)usions, the branching density is given by an arbitrary bounded non-negative Borel function, and the superprocess is characterized by a martingale problem as a di(cid:11)usion process with state space M ( R ), improving and extending considerably the construction of Wang (1997, 1998). It is then proved in a special case that a suitable rescaled process of the superprocess converges to the usual super Brownian motion. An extension to measure-valued branching catalysts is also discussed.


Introduction
For a given topological space E, let B(E) denote the totality of all bounded Borel functions on E and let C(E) denote its subset comprising of continuous functions. Let M(E) denote the space of finite Borel measures on E endowed with the topology of weak convergence. Write f, µ for f dµ. For F ∈ B(M(E)) let if the limit exists. Let δ 2 F (µ)/δµ(x)δµ(y) be defined in the same way with F replaced by (δF/δµ(y)) on the right hand side. For example, if F m,f (µ) = f, µ m for f ∈ B(E m ) and µ ∈ M(E), then where x ∈ E is the ith variable of f on the right hand side.
Now we consider the case where E = R, the one-dimensional Euclidean space. Suppose that c ∈ C(R) is Lipschitz and h ∈ C(R) is square-integrable. Let and a(x) = c(x) 2 + ρ(0) for x ∈ R. We assume in addition that ρ is twice continuously differentiable with ρ and ρ bounded, which is satisfied if h is integrable and twice continuously differentiable with h and h bounded. Then (y) µ(dx)µ(dy) (1.5) defines an operator A which acts on a subset of B(M(R)) and generates a diffusion process with state space M(R). Suppose that {W (x, t) : x ∈ R, t ≥ 0} is a Brownian sheet and {B i (t) : t ≥ 0}, i = 1, 2, · · ·, is a family of independent standard Brownian motions which are independent of {W (x, t) : x ∈ R, t ≥ 0}. By Lemma 3.1, for any initial conditions x i (0) = x i , the stochastic equations dx i (t) = c(x i (t))dB i (t) + R h(y − x i (t))W (dy, dt), t ≥ 0, i = 1, 2, · · · , (1. 6) have unique solutions {x i (t) : t ≥ 0} and, for each integer m ≥ 1, {(x 1 (t), · · · , x m (t)) : t ≥ 0} is an m-dimensional diffusion process which is generated by the differential operator (1.7) In particular, {x i (t) : t ≥ 0} is a one-dimensional diffusion process with generator G := (a(x)/2)∆. Because of the exchangebility, a diffusion process generated by G m can be regarded as an interacting particle system or a measure-valued process. Heuristically, a(·) represents the speed of the particles and ρ(·) describes the interaction between them. The diffusion process generated by A arises as the high density limit of a sequence of interacting particle systems described by (1.6); see Wang (1997Wang ( , 1998 and section 4 of this paper. For σ ∈ B(R) + , we may also define the operator B by . (1.8) A Markov process generated by L := A + B is naturally called a superprocess with dependent spatial motion (SDSM) with parameters (a, ρ, σ), where σ represents the branching density of the process. In the special case where both c and σ are constants, the SDSM was constructed in Wang (1997Wang ( , 1998) as a diffusion process in M(R ), whereR = R ∪ {∂} is the one-point compactification of R. It was also assumed in Wang (1997Wang ( , 1998) that h is a symmetric function and that the initial state of the SDSM has compact support in R.
Stochastic partial differential equations and local times associated with the SDSM were studied in Dawson et al (2000a, b).
The SDSM contains as special cases several models arising in different circumstances such as the one-dimensional super Brownian motion, the molecular diffusion with turbulent transport and some interacting diffusion systems of McKean-Vlasov type; see e.g. Chow (1976), Dawson (1994), Dawson and Vaillancourt (1995) and Kotelenez (1992Kotelenez ( , 1995. It is thus of interest to construct the SDSM under reasonably more general conditions and formulate it as a diffusion processes in M(R). This is the main purpose of the present paper. The rest of this paragraph describes the main results of the paper and gives some unsolved problems in the subject. In section 2, we define some function-valued dual process and investigate its connection to the solution of the martingale problem of a SDSM. Duality method plays an important role in the investigation. Although the SDSM could arise as high density limit of a sequence of interacting-branching particle systems with location-dependent killing density σ and binary branching distribution, the construction of such systems seems rather sophisticated and is thus avoided in this work.
In section 3, we construct the interacting-branching particle system with uniform killing density and location-dependent branching distribution, which is comparatively easier to treat. The arguments are similar to those in Wang (1998). The high density limit of the interacting-branching particle system is considered in section 4, which gives a solution of the martingale problem of the SDSM in the special case where σ ∈ C(R) + can be extended into a continuous function onR . In section 5, we use the dual process to extend the construction of the SDSM to a general bounded Borel branching density σ ∈ B(R) + . In both sections 4 and 5, we use martingale arguments to show that, if the processes are initially supported by R, they always stay in M(R), which are new results even in the special case considered in Wang (1997Wang ( , 1998. In section 6, we prove a rescaled limit theorem of the SDSM, which states that a suitable rescaled SDSM converges to the usual super Brownian motion if c(·) is bounded away from zero. This describes another situation where the super Brownian motion arises universally; see also Durrett and Perkins (1998) and Hara and Slade (2000a, b). When c(·) ≡ 0, we expect that the same rescaled limit would lead to a measure-valued diffusion process which is the high density limit of a sequence of coalescing-branching particle systems, but there is still a long way to reach a rigorous proof. It suffices to mention that not only the characterization of those high density limits but also that of the coalescing-branching particle systems themselves are still open problems. We refer the reader to Evans and Pitman (1998) and the references therein for some recent work on related models. In section 7, we consider an extension of the construction of the SDSM to the case where σ is of the form σ =η with η belonging to a large class of Radon measures on R, in the lines of Dawson andFleischmann (1991, 1992). The process is constructed only when c(·) is bounded away from zero and it can be called a SDSM with measure-valued catalysts. The transition semigroup of the SDSM with measure-valued catalysts is constructed and characterized using a measurevalued dual process. The derivation is based on some estimates of moments of the dual process. However, the existence of a diffusion realization of the SDSM with measurevalued catalysts is left as another open problem in the subject.
Notation: Recall thatR = R ∪ {∂} denotes the one-point compactification of R. Let λ m denote the Lebesgue measure on R m . Let C 2 (R m ) be the set of twice continuously differentiable functions on R m and let C 2 ∂ (R m ) be the set of functions in C 2 (R m ) which together with their derivatives up to the second order can be extended continuously toR . Let C 2 0 (R m ) be the subset of C 2 ∂ (R m ) of functions that together with their derivatives up to the second order vanish rapidly at infinity. Let (T m t ) t≥0 denote the transition semigroup of the m-dimensional standard Brownian motion and let (P m t ) t≥0 denote the transition semigroup generated by the operator G m . We shall omit the superscript m when it is one. Let (P t ) t≥0 andĜ denote the extensions of (P t ) t≥0 and G toR with ∂ as a trap. We denote the expectation by the letter of the probability measure if this is specified and simply by E if the measure is not specified.
We remark that, if |c(x)| ≥ > 0 for all x ∈ R, the semigroup (P m t ) t>0 has density p m t (x, y) which satisfies where g m t (x, y) denotes the transition density of the m-dimensional standard Brownian motion; see e.g. Friedman (1964, p.24).

Function-valued dual processes
In this section, we define a function-valued dual process and investigate its connection to the solution of the martingale problem for the SDSM. Recall the definition of the generator L := A + B given by (1.5) and (1.8) and where x m−1 is in the places of the ith and the jth variables of f on the right hand side. It follows that Let {Γ k : 1 ≤ k ≤ M 0 − 1} be a sequence of random operators which are conditionally independent given {M t : t ≥ 0} and satisfy where Φ i,j is defined by (2.3). Let B denote the topological union of {B(R m ) : m = 1, 2, · · ·} endowed with pointwise convergence on each B(R m ). Then is also a Markov process. To simplify the presentation, we shall suppress the dependence of {Y t : t ≥ 0} on σ and let E σ m,f denote the expectation given M 0 = m and Y 0 = f ∈ C(R m ), just as we are working with a canonical realization of 1 For any f ∈ B(R m ) and any integer m ≥ 1, where · denotes the supremum norm.
Proof. The left hand side of (2.8) can be decomposed as m−1 k=0 A k with Then we get the conclusion.

Lemma 2.2
Suppose that σ n → σ boundedly and pointwise and µ n → µ in M(R) as n → ∞. Then, for any f ∈ B(R m ) and any integer m ≥ 1, and g(y)σ n (y)dy → g(y)σ(y)dy by weak convergence, so that Since {µ n } is tight and {σ n } is bounded, one can easily see that {p t (x, y)µ n (dx)σ n (y)dy} is a tight sequence and hence p t (x, y)µ n (dx)σ n (y)dy → p t (x, y)µ(dx)σ(y)dy by weak convergence. Therefore, the value of (2.10) converges as n → ∞ to Applying bounded convergence theorem to (2.7) we get inductively Then the result follows from (2.7).

Theorem 2.1 Let D(L) be the set of all functions of the form
Proof. In view of (2.6), the general equality follows by bounded pointwise approximation once it is proved for In view of (2.4) we have The following calculations are guided by the relation (2.12). In the sequel, we assume that {X t : t ≥ 0} and {(M t , Y t ) : t ≥ 0} are defined on the same probability space and are independent of each other. Suppose that for each n ≥ 1 we have a partition where the last step holds by the right continuity of {X t : t ≥ 0}. Using again the independence and the martingale problem for {X t : t ≥ 0}, where we have also used the right continuity of {(M t , Y t ) : t ≥ 0} for the last step. Finally, Since the semigroups (P m t ) t≥0 are strongly Feller and strongly continuous, {Y t : t ≥ 0} is continuous in the uniform norm in each open interval between two neighboring jumps of {M t : t ≥ 0}. Using this, the left continuity of {X t : t ≥ 0} and dominated convergence, we see that the above value is equal to Combining those together we see that the value of (2.13) is in fact zero and hence (2.11) follows.

Theorem 2.2 Let D(L) be as in Theorem 2.1 and let {w
Proof. Let Q t (µ, ·) denote the distribution of w t under Q µ . By Theorem 2.1 we have (2.14). Let us assume first that σ(x) ≡ σ 0 for a constant σ 0 . In this case, Then for each f ∈ B(R) + the power series has a positive radius of convergence. By this and Billingsley (1968, p.342) it is not hard to show that Q t (µ, ·) is the unique probability measure on M(R) satisfying (2.14). Now the result follows from Ethier and Kurtz (1986, p.184). For a non-constant σ ∈ B(R) + , let σ 0 = σ and observe that . Then the power series (2.15) also has a positive radius of convergence and the result follows as in the case of a constant branching rate.

Interacting-branching particle systems
In this section, we give a formulation of the interacting-branching particle system. We first prove that equations (1.6) have unique solutions. Recall that c ∈ C(R) is Lipschitz, h ∈ C(R) is square-integrable and ρ is twice continuously differentiable with ρ and ρ bounded. The following result is an extension of Lemma 1.3 of Wang (1997) where it was assumed that c(x) ≡ const.

Lemma 3.1 For any initial conditions
Let l(c) ≥ 0 be any Lipschitz constant for c(·). By a martingale inequality we have Using the above inequality inductively we get which is clearly the unique solution of (1.6). It is easy to Because of the exchangebility, the G m -diffusion can be regarded as a measure-valued Markov process. Let N(R) denote the space of integer-valued measures on R. For θ > 0,

Lemma 3.2 For any integers m, n ≥ 1 and any
Proof. By (3.1), we have On the other hand, for 1 ≤ i = j ≤ m, where {· · ·} = { for all 1 ≤ l 1 , · · · , l n ≤ m with l α = i and l β = j}. It follows that

Using this and (3.4) with c(x
Then we have the desired result from (3.4) and (3.5).
Suppose that X(t) = (x 1 (t), · · · , x m (t)) is a Markov process in R m generated by G m .
Based on (1.2) and Lemma 3.2, it is easy to show that ζ(X(t)) is a Markov process in M θ (R) with generator A θ given by In particular, if for f ∈ C 2 (R n ) and {φ i } ⊂ C 2 (R), then Now we introduce a branching mechanism to the interacting particle system. Suppose that for each x ∈ R we have a discrete probability distribution p(x) = {p i (x) : i = 0, 1, · · ·} such that each p i (·) is a Borel measurable function on R. This serves as the distribution of the offspring number produced by a particle that dies at site x ∈ R. We assume that and where µ ∈ M θ (R) is given by For a constant γ > 0, we define the bounded operator B θ on B(M θ (R)) by In view of (1.6), A θ generates a Feller Markov process on M θ (R), then so does L θ := A θ + B θ by Ethier-Kurtz (1986, p.37). We shall call the process generated by L θ an interactingbranching particle system with parameters (a, ρ, γ, p) and unit mass 1/θ. Heuristically, each particle in the system has mass 1/θ, a(·) represents the migration speed of the particles and ρ(·) describes the interaction between them. The branching times of the system are determined by the killing density γθ 2 [θ ∧µ (1)], where the truncation "θ ∧µ (1)" is introduced to make the branching not too fast even when the total mass is large. At each branching time, with equal probability, one particle in the system is randomly chosen, which is killed at its site x ∈ R and the offspring are produced at x ∈ R according to the for some constant 0 < ξ j < (j − 1)/θ. This follows from (3.11) and (3.12) by Taylor's expansion.

Continuous branching density
In this section, we shall construct a solution of the martingale problem of the SDSM with continuous branching density by using particle system approximation. Assume that σ ∈ C(R) can be extended continuously toR . Let A and B be given by (1.5) and (1.8), respectively. Observe that, if for f ∈ C 2 (R n ) and {φ i } ⊂ C 2 (R), then : t ≥ 0} is a sequence of cádlág interacting-branching particle systems with parameters (a, ρ, γ k , p (k) ), unit mass 1/θ k and initial states X In an obvious way, we may also regard {X

Proof. By the assumption (3.9), it is easy to show that
: t ≥ 0} satisfies the compact containment condition of Ethier and Kurtz (1986, p.142). Let L k denote the generator of {X (k) t : t ≥ 0} and let F be given by (4.1) with f ∈ C 2 0 (R n ) and with each φ i ∈ C 2 ∂ (R) bounded away from zero. Then is a martingale and the desired tightness follows from the result of Ethier and Kurtz (1986, p.145).

Lemma 4.2 Let D(L) be the totality of all functions of the form (4.1) with f ∈ C 2
0 (R n ) and with each φ i ∈ C 2 ∂ (R) bounded away from zero. Suppose further that γ k σ k → σ uniformly and µ k → µ ∈ M(R ) as k → ∞. Then any limit point Q µ of the distributions of {X Proof. We use the notation introduced in the proof of Lemma 4.1. By passing to a subsequence if it is necessary, we may assume that the distribution of {X (k) t : t ≥ 0} on D([0, ∞), M(R )) converges to Q µ . Using Skorokhod's representation, we may assume that the processes {X (k) t : t ≥ 0} are defined on the same probability space and the sequence converges almost surely to a cádlág process {X t : t ≥ 0} with distribution Q µ on D([0, ∞), M(R )); see e.g. Ethier and Kurtz (1986, p.102). Let K(X) = {t ≥ 0 : P {X t = X t − } = 1}. By Ethier and Kurtz (1986, p.118), for each t ∈ K(X) we have a.s. lim k→∞ X (k) t = X t . Recall that f and f ij are rapidly decreasing and each φ i is bounded away from zero. Since γ k a k → σ uniformly, for t ∈ K(X) we have a.s. lim k→∞ L k F (X . By Ethier and Kurtz (1986, p.31), the set K(X) is at most countable. Then By the right continuity of {X t : t ≥ 0}, the equality is a martingale. As in Wang (1998, pp.783-784) one can show that {X t : t ≥ 0} is in fact a.s. continuous.

Lemma 4.3 Let D(L) be as in Lemma 4.2. Then for each µ ∈ M(R ), there is a probability measure Q µ on C([0, ∞), M(R )) under which (4.4) is a martingale for each F ∈ D(L).
Proof. It is easy to find µ k ∈ M θ k (R) such that µ k → µ as k → ∞. Then, by Lemma 4.2, it suffices to construct a sequence (γ k , p (k) ) such that γ k σ k → σ as k → ∞. This is elementary. One choice is described as follows.

Lemma 4.4 Let Q µ be given by Lemma 4.3. Then for n ≥ 1, t ≥ 0 and µ ∈ M(R) we have
is a martingale, we get Then the desired estimate follows by Fatou's Lemma. The last assertion is an immediate consequence of Lemma 4.3.

Lemma 4.5 Let Q µ be given by Lemma 4.3. Then for µ ∈ M(R) and
is a Q µ -martingale with quadratic variation process Proof. It is easy to check that, if F n (µ) = φ, µ n , then It follows that both (4.5) and are martingales. By (4.5) and Itô's formula we have Comparing (4.7) and (4.8) we get the conclusion.
Observe that the martingales {M(φ) : t ≥ 0} defined by (4.5) form a system which is linear in φ ∈ C 2 ∂ (R). Because of the presence of the derivative φ in the variation process (4.6), it seems hard to extend the definition of {M(φ) : t ≥ 0} to a general function φ ∈ B(R ). However, following the method of Walsh (1986), one can still define the stochastic integral

Proof. For any partition
Using Lemma 4.5 we have Combining those we get the desired conclusion. The desired result will follow once it is proved that For any φ ∈ C 2 ∂ (R), we may use Lemma 4.6 to see that is a continuous martingale with quadratic variation process By a martingale inequality we have Replacing φ by φ k in the above and letting k → ∞ we obtain (4.9).
Combining Theorems 2.2 and 4.1 we get the existence of the SDSM in the case where σ ∈ C(R) + extends continuously toR .

Measurable branching density
In this section, we shall use the dual process to extend the construction of the SDSM to a general bounded Borel branching density. Given σ ∈ B(R) + , let {(M t , Y t ) : t ≥ 0} be defined as in section 2. Choose any sequence of functions {σ k } ⊂ C(R) + which extends continuously toR and σ k → σ boundedly and pointwise. Suppose that {µ k } ⊂ M(R) and µ k → µ ∈ M(R) as k → ∞. For each k ≥ 1, let {X (k) t : t ≥ 0} be a SDSM with parameters (a, ρ, σ k ) and initial state µ k ∈ M(R) and let Q k denote the distribution of {X  : t ≥ 0} and let F be given by (4.1) with f ∈ C 2 0 (R n ) and with {φ i } ⊂ C 2 ∂ (R). Then is a martingale. Since the sequence {σ k } is uniformly bounded, the tightness of {X (k) t : t ≥ 0} in C([0, ∞), M(R )) follows from Lemma 4.4 and the result of Ethier and Kurtz (1986, p.145). We shall prove that any limit point of {Q k } is supported by C([0, ∞), M(R)) so that {Q k } is also tight as probability measures on C ([0, ∞), M(R)). Without loss of generality, we may assume Q k converges as k → ∞ to Q µ by weak convergence of probability measures on C ([0, ∞), M(R )). Let φ n ∈ C 2 (R) + be such that φ n (x) = 0 when x ≤ n and φ n (x) = 1 when x ≥ 2n and φ n → 0 as n → ∞. Fix u > 0 and let m n be such that φ mn (x) ≤ 2P t φ n (x) for all 0 ≤ t ≤ u and x ∈ R. For any α > 0, the paths w ∈ C([0, ∞), M(R )) satisfying sup 0≤t≤u φ mn , w t > α constitute an open subset of C([0, ∞), M(R )). Then, by an equivalent condition for weak convergence, As in the proof of Theorem 4.1, one can see that the right hand side goes to zero as n → ∞. Then Q µ is supported by C([0, ∞), M(R)).
on M(R) converges as k → ∞ to a probability measure Q t (µ, ·) on M(R) given by to a smaller subsequence {k i } we may assume that f, ν m Q (k i ) t (µ k i , dν) converges to a finite measure K t (µ, dν) on M(R ). Then we must have K t (µ, dν) = f, ν m Q t (µ, dν). By Lemma 2.2 and the proof of Theorem 2.2, Q t (µ, ·) is uniquely determined by (5.1). Therefore, Q we have the Chapman-Kolmogorov equation.
The existence of a SDSM with a general bounded measurable branching density function σ ∈ B(R) is given by the following Proof. Let Q µ be the limit point of any subsequence {Q k i } of {Q k }. Using Skorokhod's representation, we may construct processes {X Ethier and Kurtz (1986, p.102). For any {H j } n+1 j=1 ⊂ C(M(R )) and 0 ≤ t 1 < · · · < t n < t n+1 we may use Theorem 5.1 and dominated convergence to see that Then {X t : t ≥ 0} is a Markov process with transition semigroup (Q t ) t≥0 and actually Q k → Q µ as k → ∞. The strong Markov property holds since (Q t ) t≥0 is Feller by (5.1). To see the last assertion, one may simply check that (L, D(L)) is a restriction of the generator of (Q t ) t≥0 .

Rescaled limits
In this section, we study the rescaled limits of the SDSM constructed in the last section. Given any θ > 0, we defined the operator Proof. We shall compute the generator of {X θ t : t ≥ 0}. Let F (µ) = f ( φ, µ ) with f ∈ C 2 (R) and φ ∈ C 2 (R). Note that F • K θ (µ) = F (K θ µ) = f ( φ 1/θ , µ ). By the theory of transformations of Markov processes, {K θ X t : t ≥ 0} has generator L θ such that L θ F (µ) = L(F • K θ )(K 1/θ µ). Since it is easy to check that Then one may see that {θ −2 K θ X t : t ≥ 0} has generator L θ such that Then it is easy to check that for each T > 0, so there is a λ × λ × Q µ -measurable function X t (ω, x) satisfying (6.1) and and by Schwarz inequality, By this and (6.2) we get On the other hand, using (2.8) and (5.1) one may see that Then letting r → 0 in (6.3) we have completing the proof.
converges as θ → ∞ to that of a super Brownian motion with underlying generator (a ∂ /2)∆ and uniform branching density σ ∂ .
Proof. Since σ θ = σ and X θ 0 = µ, as in the proof of Lemma 5.1 one can see that the family {X θ t : t ≥ 0} is tight in C([0, ∞), M(R)). Choose any sequence θ k → ∞ such that the distribution of {X θ k t : t ≥ 0} converges to some probability measure Q µ on C([0, ∞), M(R)). We shall prove that Q µ is the solution of the martingale problem for the super Brownian motion so that actually the distribution of {X θ t : t ≥ 0} converges to Q µ as θ → ∞. By Skorokhod's representation, we can construct processes {X with f ∈ C 2 (R) and φ ∈ C 2 (R). Then for each k ≥ 0, is a martingale, where L k is given by Observe that Then we have In the same way, one sees that Using the density process of {X (k) t : t ≥ 0} we have the following estimates By (6.4), for any fixed t ≥ 0, is uniformly bounded in k ≥ 1. Since ρ θ k (x − y) → 0 for λ × λ-a.e. (x, y) ∈ R 2 and since Using (6.6),(6.7), (6.8) and the martingale property of (6.5) ones sees in a similar way as in the proof of Lemma 4.2 that is a martingale, where L 0 is given by This clearly implies that {X

Measure-valued catalysts
In this section, we assume |c(x)| ≥ > 0 for all x ∈ R and give construction for a class of SDSM with measure-valued catalysts. We start from the construction of a class of measure-valued dual processes. Let M B (R) denote the space of Radon measures ζ on R to which there correspond constants b(ζ) > 0 and l(ζ) > 0 such that x ∈ R. where h( , ζ; t) = const · b(ζ) 2l(ζ) + √ 2π t , t >0.
Proof. Using (1.9) and (7.1) we have giving the desired inequality.
Returning to the decomposition we get the desired estimate. Proof. Based on (7.7), the desired result follows by a similar argument as in the proof of Lemma 2.2.
Let η ∈ M B (R) and let η k be defined as in Lemma 7.2. Let σ k denote the density of η k with respect to the Lebesgue measure and let {X (k) t : t ≥ 0} be a SDSM with parameters (a, ρ, σ k ) and initial state µ k ∈ M(R). Assume that µ k → µ weakly as k → ∞. Then we have the following Proof. With Lemmas 7.3 and 7.4, this is similar to the proof of Theorem 5.1.
A Markov process with transition semigroup defined by (7.10) is the so-called SDSM with measure-valued catalysts.