STATE CLASSIFICATION FOR A CLASS OF INTERACTING SUPERPROCESSES WITH LOCATION DEPENDENT BRANCHING

The spatial structure of a class of superprocesses which arise as limits in distribution of a class of interacting particle systems with location dependent branching is investigated. The criterion of their state classiﬂcation is obtained. Their eﬁective state space is contained in the set of purely-atomic measures or the set of absolutely continuous measures according as one diﬁusive coe–cient c ( x ) · 0 or j c ( x ) j ‚ " > 0 while another diﬁusive coe–cient h 2 C 2 b ( R )


Introduction and main result
In Dawson-Li-Wang [3], a class of interacting branching particle systems with location dependent branching, which generalizes the model introduced in Wang [14], is introduced and the limiting superprocesses, which will be called superprocesses with dependent spatial motion and branching (SDSBs), are constructed and characterized. In Theorem 6.1 of [3], it is proved that when the motion coefficient satisfies uniformly elliptic condition (which means |c(x)| ≥ ε > 0 in the following model), the effective state space of the SDSBs is contained in the space of all measures which are absolutely continuous with respect to the Lebesgue measure on R. It leaves an open problem whether the effective state space of the SDSBs is contained in the space of purely-atomic measures when the motion coefficient is degenerate (which means c(x) ≡ 0 in the following model). In our model, the motions of the particles are not independent. This can be seen from their non-zero quadratic variation processes. This is one essential difference from the Super-Brownian motion. Another essential difference is that the branching coefficient in our model depends on the spatial location. Therefore, motion of the particles affects the branching. This is a new class of interaction. To compare with other existing models, reader is referred to [1], [2], [4], [7], [8], [9], [10], to name only a few. In the present paper, the spatial structure of the SDSBs is investigated. We will give solution to above mentioned open problem left in [3]. Combining with the result proved in [3], we will give a criterion of state classification for SDSBs. Before introducing our model, let us give some notations. Let R = (−∞, ∞),R = R ∪ {∂}, the one-point compactification of R, N = {1, 2, 3, · · · },N = {0} ∪ N, C(R) be the space of all continuous functions on R, C b (R) be the space of all bounded continuous functions on R, C 0 (R) be the space of all continuous functions vanishing at infinity, C L (R) be the space of all Lipschitz continuous functions on R, and C n b (R) be the space of all the functions which has bounded, continuous derivatives up until and including order n. Now, let us introduce our model. Suppose that {W (x, t) : x ∈ R, t ≥ 0} is a Brownian sheet (see [12]) and {B i (t) : t ≥ 0},i ∈ N, is a family of independent standard Brownian motions which are independent of {W (x, t) : x ∈ R, t ≥ 0}. For each natural number n which serves as a control parameter for our finite branching particle systems, we consider a system of particles (initially, there are m n 0 particles) which move, die and produce offspring in a random medium on R. The diffusive part of such a branching particle system has the form where c ∈ C L (R) and h ∈ C 2 b (R) is a square-integrable function. By Lemma 3.1 of [3], for any initial conditions x n i (0) = x i , the stochastic equations (1.1) have unique strong solution {x n i (t) : t ≥ 0} and, for each integer m ≥ 1, {(x n 1 (t), · · · , x n m (t)) : t ≥ 0} is an m-dimensional diffusion process which is generated by the differential operator In particular, {x n i (t) : t ≥ 0} is a one-dimensional diffusion process with generator G := (a(x)/2)∆, where ∆ is the Laplacian operator, and a(x) := c 2 (x) + ρ(0) for x ∈ R. The function ρ is twice continuously differentiable with ρ and ρ bounded since h is integrable and twice continuously differentiable with h and h bounded. The quadratic variational process for the system given by (1.1) is where we set δ {i=j} = 1 or 0 according as i = j or i = j, where i, j ∈ N. Here x n i (t) is the location of the i th particle. We assume that each particle has mass 1/θ n and branches at rate γθ n , where γ ≥ 0 and θ ≥ 2 are fixed constants. We assume that when a particle 1 θ n δ x , which has location at x, dies, it produces k particles with probability p k (x); x ∈ R, k ∈N. This means that the branching mechanism depends on the spatial location. The offspring distribution is assumed to satisfy: (1.5) The second condition indicates that we are solely interested in the critical case. After branching, the resulting set of particles evolve in the same way as their parent and they start off from the parent particle's branching site. Let m n t denote the total number of particles at time t . Denote the empirical measure process by In order to obtain measure-valued processes by use of an appropriate rescaling, we assume that there is a positive constant ξ > 0 such that m n 0 /θ n ≤ ξ for all n ≥ 0 and that weak convergence of the initial laws µ n 0 ⇒μ holds, for some finite measureμ. As for the convergence from branching particle systems to a SDSB, reader is referred to [14] and [3]. Let E := M (R) be the Polish space of all bounded Radon measures on R with the weak topology defined by By Ito's formula and the conditional independence of motions and branching, we can obtain the following formal generators (usually called pregenerators) for the limiting measure-valued processes: and We cite two theorems proved in [3].
be a square-integrable function on R, and σ(x) ∈ B(R) + . Then, for any µ ∈ E, (L c,σ , δ µ )-martingale problem (MP) has a unique solution which is a diffusion process.
Proof: For the proof of this theorem, reader is referred to the section 5 of [3].
where L is the Lebesgue measure on R and µ t < < L means that µ t is absolutely continuous with respect to Lebesgue measure L on R.
Proof: For the proof of this theorem, reader is referred to the Theorem 6.1 of [3].
We have following main result: Then, for any t > 0, µ σ t is a purely-atomic measure. Furthermore, for any given t 0 > 0 and conditioned on µ σ t has following representation: where W (y, u) is a Brownian sheet and {B i (t) : i ≥ 1} are a sequence of independent onedimensional Brownian motions which are independent of W (y, u), {I(t) ⊂ N : t > t 0 } is no-increasing random subsets in t in terms of set inclusion order.

Proof of the main result
The strategies to prove our main result can be described as follows: (1) Generator Decomposition Technique : We decompose the branching generator as follows: where the operators are define by (1.8), (2.14), and (2.21), respectively. By virtue of B ε/2 and existing results of [13], this decomposition technique helps us to prove and explain that our concerned interacting superprocesses with a variable coefficient branching generator immediately enters into the purely-atomic measure valued state even if the initial state is an absolutely continuous measure.
(2) Branching Mechanism Reconstruction: By reconstruction of the branching mechanism, the variable coefficient σ(x) of the branching generator is transformed as a location dependent branching rate with location independent, equal probability binary branching in the branching particle model. If inf x∈R σ(x) ≥ ε > 0, then the mean life time of the particles with variable branching coefficient σ(x) is shorter than that of the particles with constant branching coefficient ε.
(3) Trotter's Product Formula: Based on the branching mechanism reconstruction, we will use Trotter's product formula and a dominating method, which shows that the number of particles of a purely-atomic measure-valued superprocess with a variable branching coefficient σ(x) is dominated by the number of particles of a purely-atomic measure-valued superprocess with a constant branching coefficient ε if σ(x) ≥ ε > 0, to reach our conclusion.
To prove our main result, we need two lemmas. (2.14) Then, for any t 0 ≥ 0 and for any µ t0 ∈ E, (L 0,d , δ µt 0 )-MP has a unique solution {µ t : t ≥ t 0 ≥ 0} which has sample paths in C([t 0 , ∞), E). If the initial state is given by µ t0 = i∈I(t0) a i (t 0 )δ xi(t0) which is a purely-atomic measure with x i (t 0 ) = x j (t 0 ) if i = j and i, j ∈ I(t 0 ), where I(t 0 ) is at most a countable set, then for any t ∈ [t 0 , ∞), µ t has following representation:

15)
and x i (t) satisfies

16)
where W (y, u) is a Brownian sheet and {B i (t) : i ∈ N} are a sequence of independent onedimensional Brownian motions which are independent of W (y, u).
Proof: In order to simplify the notation, without loss of generality in the following we simply assume that t 0 = 0. The existence, uniqueness of the (L 0,d , δ µ0 )-MP, and its solution being a diffusion process follow from Theorem 1.1 with c(x) ≡ 0. We will use Itô's formula to prove the remaining parts of the lemma. Suppose that W (y, u) is a Brownian sheet and {B i (t) : i ∈ N, t ≥ 0} are a sequence of independent one-dimensional Brownian motions which are independent of W (y, u). Let {a i (t)} be the unique solution of (2.15) and {x i (t)} be the unique solution of (2.16) with t 0 = 0. Define µ t = i∈I(t) a i (t)δ xi(t) . Since h ∈ C 2 b (R) is a square-integrable function on R and c ≡ 0, according to the behavior of the generator A 0 , the location processes {x i (t) : t ≥ 0, i ∈ I(0)} have following coalescence property (See section 1.2 and the proof of Lemma 1.2 in [13]).
Coalescence Property: A branching particle system is said to have coalescence property if the particle location processes are diffusion processes and for any two particles either they never separate or they never meet according as they start off from same initial location or not.
Proof: The conclusion that for any µ 0 , ν 0 ∈ E, (L 0,d , δ µ0 )-MP ((L 0,ε/2 , δ ν0 )-MP) has a unique solution {µ t : t ≥ 0} ({ν t : t ≥ 0}) which has sample paths in C([0, ∞), E) is proved in [3]. The purely-atomic representation is proved by above lemma 2.1. To complete the proof, it only needs to prove that P(τ a ≤ τ b ) = 1. In order to prove this result, we will compare two operators L 0,d and L 0,ε/2 . We will use different point of view to explain the behaviors of B d and B ε/2 . At the beginning of this paper, we have introduced our model of interacting branching particle systems, where σ(x) = γ(m 2 (x) − 1), γ is a constant branching rate and the offspring distribution depends on spatial location. Now we remodel the interacting branching particle systems. For both the operator L 0,d and the operator L 0,ε/2 , their corresponding interacting branching systems can be alternatively described as follows: For each n which serves as a control parameter for a finite branching particle system, we consider a system of particles (initially, there are m n 0 particles) which move, die and produce offspring in a random medium on R. The diffusive part of such a branching particle system has the form where W (y, t) is a Brownian sheet and h ∈ C 2 b (R) is a square-integrable function. Here x n i (t) is the location of the i th particle. The branching mechanisms for the operator L 0,d and the operator L 0,ε/2 are different.
(1) For the operator L 0,d , we assume that each particle has mass 1/θ n and branches at ratẽ γ(x)θ n if the particle's current location is x, where θ ≥ 2 is a fixed constant andγ(x) := σ(x) − ε/2 ≥ ε/2 > 0. We assume that all particles undergo binary branching with equal probability 1 2 or more precisely after a particle dies, it is replaced by 0 or 2 particles of same kind with equal probability 1 2 . Thus, the offspring distribution is independent of spatial location. Therefore, in other words, this says that particles undergo binary branching with equal probability 1 2 and each particle's lifetime is measured by a clock whose speed changes as this particle's location changes. By Itô's formula, it is not difficult to find that the pregenerator of the limiting superprocess of the interacting branching particle systems is L 0,d . In [3], it is proved that the martingale problem for L 0,d is well-posed.
(2) For the operator L 0,ε/2 , we assume that each particle has mass 1/θ n and branches at rate (ε/2)θ n which is independent of the particle's current location x, where θ ≥ 2 and ε are two fixed constants. We assume that all particles undergo binary branching with equal probability 1 2 . Thus, the offspring distribution is independent of spatial location. Therefore, this means that a particle's lifetime is measured by a clock whose speed is fixed. After this particle dies, it is replaced by 0 or 2 particles with equal probability 1 2 . By Itô's formula, it is not difficult to find that the pregenerator of the limiting superprocess of the interacting branching particle systems is L 0,ε/2 . In [14] or [3] it is already proved that the martingale problem for L 0,ε/2 is well-posed. Based on above reconstruction of the models of the interacting branching particle systems, we have following comparison for a(t)δ x(t) and b(t)δ x(t) , the unique solutions of (L 0,d , δ (a(0)δ x(0) ) )-MP and (L 0,ε/2 , δ (a(0)δ x(0) ) )-MP, respectively. First, they have same location trajectory x(t). Second, they have same binary branching mechanism. Third, the only difference is their mass processes. This difference is produced by their different branching rates. Since inf x∈R (σ(x) − ε/2) ≥ ε/2, the continuous branching process a(t)'s clock speed is uniformly quicker than or equal to that of the continuous branching process b(t). Thus, P(τ a ≤ τ b ) = 1 holds.
Proof of Theorem 1.3: For any measure µ 0 ∈ E, let {γ(t) : t ≥ 0} be the unique solution to the (B ε/2 , δ µ0 )-MP with sample paths in C([0, ∞), E) on a probability space (Ω, F, P). Then, by Theorem 1.1 of [13], for any t > 0, γ(t) is a purely-atomic-measure (This is just the mutation-free Fleming-Viot process, see [6] [5]). For any natural integer k, we assume that at t/k, γ(t/k) can be represented as follows: where I(t/k) is at most a countable set such that x i (0) = x j (0) if i = j and i, j ∈ I(t/k), and c i (t/k) > 0 for all i ∈ I(t/k). Let {T ε/2 t } be the Feller semigroup generated by B ε/2 , {U d t } be the Feller semigroup generated by L 0,d , and {U σ t } be the Feller semigroup generated by L 0,σ . By Trotter's product formula (See [11]), for any F ∈ C 0 (E), we have for all t ≥ 0, uniformly on bounded intervals. Define Let {V ε/2 t } be the Feller semigroup generated by L 0,ε/2 , and {V ε t } be the Feller semigroup generated by L 0,ε . By Trotter's product formula (See [11]), for any F ∈ C 0 (E), we have for all t ≥ 0, uniformly on bounded intervals. Let {µ σ t : t ≥ 0} be the unique solution to the (L 0,σ , δ µ0 )-MP with µ 0 ∈ E and Let {ν ε t : t ≥ 0} be the unique solution to the (L 0,ε , δ µ0 )-MP with same µ 0 ∈ E. We already proved in [13] that {ν ε t : t > 0} is a purely-atomic measure valued process. Now we want to prove that for any t > 0, µ σ t is also a purely-atomic measure and the number of atoms of µ σ t is at most equal to that of ν ε t . To this end, we will construct the stochastic representations for both [ t/k ] k conditionally on γ(t/k). On the same probability space (Ω, F, P), let W (x, t) be a Brownian sheet and {B i (t) : i ∈ N} be a sequence of independent one-dimensional Brownian motions which are independent of W (x, t). Conditioned on γ(t/k), for each i ∈ I(t/k) we construct following sequences: For the location processes, define and for 1 ≤ m ≤ 2k definẽ (2.30) For the mass processes, definec i (t/k) := c i (t/k) and (2.31) c i ((2j + 1)t/k) :=c i (2jt/k) + where I(mt/k) is a random subset of I(t/k) such thatc i (mt/k) > 0 if i ∈ I(mt/k). For any function F ∈ C(E) and any natural integer l satisfying 1 ≤ l ≤ k , by (2.29),(2.31), and Lemma 2.1, we can get  where I (mt/k) is a random subset of I(t/k) such thatb i (mt/k) > 0 if i ∈ I (mt/k). By (2.29),(2.37), and Lemma 2.1 with σ(x) ≡ ε, for any 1 ≤ l ≤ k we can get V ε,k t/k F (ν ε,k (2l−1)t/k ) = E ν ε,k (2l−1)t/k F (ν ε,k (2l)t/k ) (2.39) Thus, we have (2.40) By Lemma 2.2, we know that for any natural integer k and for any 1 ≤ m ≤ 2k, I(mt/k) ⊆ I (mt/k) holds almost surely with respect to P. Thus, this is true in distribution for the limiting processes {µ σ t : t ≥ 0} and {ν ε t : t ≥ 0} and we conclude that for any t > 0, µ σ t is a purely-atomic-measure. The remaining conclusion follows from Itô's formula.