LONG-TERM BEHAVIOR FOR SUPERPROCESSES OVER A STOCHASTIC FLOW

We study the limit of a superprocess controlled by a stochastic (cid:176)ow as t ! 1 . It is proved that when d • 2, this process suﬁers long-time local extinction; when d ‚ 3, it has a limit which is persistent. The stochastic log-Laplace equation conjectured by Skoulakis and Adler [7], and studied by this author [12], plays a key role in the proofs, similar to the one played by the log-Laplace equation in deriving long-term behavior for the standard super-Brownian motion.


Introduction and main results
Suppose that a branching system is affected by a Brownian motion W (t) which applies to every individual in that system. Between branchings, the motion of the ith particle is governed by an individual Brownian motion B i (t) and the common Brownian motion W (t): dη i (t) = b(η i (t))dt + σ 1 (η i (t))dW (t) + σ 2 (η i (t))dB i (t) where b : R d → R d , σ 1 , σ 2 : R d → R d×d are measurable functions, W, B 1 , B 2 , · · · are independent d-dimensional Brownian motions. Each individual, independent of others, splits into 2 or dies with equal probabilities after its standard exponential time runs out. This system has been constructed by Skoulakis and Adler [7] (a similar model has been investigated by Wang [9] and Dawson et al [2]). It is indicated by [7] that there are situations in which a common background noise would be a natural effect to include in a stochastic model. In fact, it can be regarded as an outside force which applies to each individual of the system. Because of the introduction of this outside force, the process no longer has the multiplicative property which is the key to the successes in the study of the classical superprocesses. To overcome this difficulty, new tools have to be developed. The aim of this paper is to study the long-term behavior of this process.
Let M F (R d ) be the collection of all finite Borel measures on R d . Let C 2 0 (R d ) be the collection of functions of compact support and continuous derivatives up to order 2. Let C 2 0 (R d ) + consist of the nonnegative elements of C 2 0 (R d ). It has been established by Skoulakis and Adler [7] that the scaling limit of the system is an M F (R d )-valued superprocess X t which is uniquely characterized by the following martingale problem: X 0 = µ ∈ M F (R d ) and for any φ ∈ C 2 0 (R d ), is a continuous martingale with quadratic variation process 1 is the transpose of the matrix σ 1 , ∇ = (∂ 1 , · · · , ∂ d ) T is the gradient operator and µ, f represents the integral of the function f with respect to the measure µ. It was conjectured in [7] that the conditional log-Laplace transform of X t should be the unique solution to a nonlinear stochastic partial differential equation (SPDE). Namely and wheredW (r) represents the backward Itô integral: where ∆ = {r 0 , r 1 , · · · , r n } is a partition of [s, t] and |∆| is the maximum length of the subintervals. This conjecture was confirmed by Xiong [12] under the following conditions (BC) which will be assumed throughout this paper: f ≥ 0, b, σ 1 , σ 2 are bounded with bounded first and second derivatives. σ T 2 σ 2 is uniformly positive definite, σ 1 has third continuous bounded derivatives. f is of compact support. We have proved in Theorem 1.2 in [12] that (1.4) has a unique L 2 (R d ) + -valued solution in the following sense: where L * is the dual operator of L given by Further, we have shown that (cf. Lemma 2.5 in [12]) where ∂ x y r,t is the weak derivative. This then implies that for fixed r and t, y r,t (x) is a continuous function of x. Furthermore, by Lemma 2.2 in [12], we see that |y r,t (x)| is bounded by f ∞ , the supremum of f . Theorem 1.4 in [12] implies (1.3). As a consequence, we see that y s,t in (1.4) is nonnegative since −y s,t is the logarithm of a conditional Laplace transform of a nonnegative random variable. Note that in the study of the classical superprocess, the PDE satisfied by the log-Laplace transform played an important role. In this note, we shall demonstrate that the stochastic log-Laplace equation (1.4) plays a similar role in the study of the long-term behavior of the superprocess over a stochastic flow. The main idea is to show that Ee − µ,y0,t has a limit by making use of (1.4) (see also (3.3)).
If the initial measure is finite, then the total mass of X t is Feller's branching diffusion which reaches 0 in finite time. To obtain an interesting long-time limit, we need to consider the infinite measure case. In Section 3, we construct the process in the state space of tempered measures by making use of the conditional branching property of this process which is implied from the conditional log-Laplace formula (1.3). Throughout this paper, we shall assume that the initial measure µ is infinite. This article is organized as follows: In Section 2, we consider a diffusion process driven by two Brownian motions. We shall prove that, given one of the Brownian motions, the conditional process is still a Markov process. Then, we give sufficient conditions for a σ-finite measure to be invariant for this conditional process with any realization of the given Brownian motion.
In Section 3 we prove that X t converges in law to a persistent distribution when the spatial dimension d ≥ 3. In Section 4, we show that the process becomes extinct locally (eventually) when d ≤ 2. The results of this paper (Theorems 10 and 11) are analogous to the corresponding classical results for super-Brownian motion. Although the proofs are adapted from the classical ones (cf. [10], [1]), the novelty of this article is its employment of the stochastic log-Laplace equation. Furthermore, as we point out in Remark 5, the σ-finite invariant measure is not unique. Therefore, even in the classical superprocess case, the long-term limit is not unique. Throughout this paper, we use c to represent a constant which can vary from place to place. We use ξ t and ξ(t) to denote the same process whenever it is convenient to do so.

Conditional Markov processes and their infinite invariant measures
Let ξ(t) be the diffusion process given by In this section, we consider the conditional process of ξ(t) with given W . More specifically, we give sufficient conditions for an infinite measure to be invariant for this conditional process with any given W (cf. (2.5)). The existence of such a measure is crucial in next section. In Proposition 3 we give sufficient conditions for the existence of such invariant measures. In Remark 4, we give examples where such conditions are satisfied. Let E W denote the conditional expectation with W given. Let Lemma 1 ξ(t) is a conditional Markov process in the following sense: ∀ s < t and f ∈ C b (R d ), has a unique strong solution, we see that ξ(t) is a function of ξ(s), W s,t and B s,t 1 . Namely is simply the expectation of G(s, t, ξ(s), W s,t , B s,t 1 ) for B s,t 1 with ξ(s) and W s,t being fixed. Namely, it is a function of ξ(s) and W s,t , say g(s, t, ξ(s), W s,t ). Therefore, we can continue Similarly, we can show that E W (f (ξ(t))|ξ(s)) = g(s, t, ξ(s), W s,t ). (2.4) The conclusion of the lemma then follows from (2.3) and (2.4).
Given W , denote the conditional transition function by Throughout this paper, we assume that µ is an invariant measure of ξ(t): ∀ s < t, for almost all given W , Note that for t > 0 fixed, we can choose a version of g which is continuous in s < t. In fact, it can be proved that g satisfies an SPDE similar to (1.4) without the quadratic term which corresponding to the branching there. Therefore, we may and will take a version of p W such that (with t > 0 fixed) for almost all W , (2.5) holds for all s < t.
Since the condition (2.5) is not easy to verify, we seek sufficient conditions for it to hold. To this end, we write (2.1) in Stratonovich form: Intuitively, µ is an invariant measure for ξ(t) with each given realization of W if and only if it is invariant for both parts of (2.6). Namely, it should be invariant for the diffusion process and, formally, for the dynamical systeṁ If µ is finite, it is well-known (cf. Varadhan [8], and Ethier and Kurtz [3], Theorem 9.17) that µ is invariant for η(t) if and only if µ is absolutely continuous with respect to Lebesgue measure andL * µ = 0 (denote the Radon-Nickodym derivative by the same notation as the original measure), whereL * is the dual operator ofL given bȳ Under suitable conditions, it was proved in Xiong [13] that the same statement is true for µ being a σ-finite measure. Formally, the second part leads to ∇(σ T 1 µ) = 0. Therefore, we conjecture that under a suitable growth condition, µ is a σ-finite invariant measure for p W for each W if and only ifL * µ = 0 and ∇(σ T 1 µ) = 0. To investigate this conjecture, we need to study the Wong-Zakai approximation ξ (t) for the process ξ(t): Lemma 2 For any c 1 > 0, there exists a constant c = c(t) such that for any > 0, Proof: Note that (2.7) By the martingale representation theorem, there is a real-valued Brownian motion B such that It is well-known that for any K 1 > 0 and T > 0, Therefore, Now we consider t 0 σ 1 (ξ (s))Ẇ s ds. To simplify the notation, we take d = 1. Let π (s) = k for k ≤ s < (k + 1) . By Itô's formula, we have t 0 (σ 1 (ξ (s)) − σ 1 (ξ (π (s))))Ẇ s ds As we see that t 0 (σ 1 (ξ (s)) − σ 1 (ξ (π (s))))Ẇ s ds ≤ c. (2.9) As t 0 σ 1 (ξ (π (s))))Ẇ s ds = t 0 σ 1 (ξ (π (s))))dW s , similar to (2.8), we have E exp 2c 1 t 0 σ 1 (ξ (π (s))))Ẇ s ds < ∞. (2.10) The conclusion of the lemma then follows from (2.7-2.10) and a simple Cauchy-Schwarz argument.
The following proposition proves the sufficiency of the conditions in our conjecture. It remains open whether these conditions are necessary.

Proposition 3
Suppose that µ is a nonnegative function and is of derivatives up to order 2 on R d such that |∇ log µ(x)| ≤ K(1 + |x|), ∀x ∈ R d .
Proof: Let p W (s, x; t, ·) be the transition probabilities of the Markov process ξ (t) with given W . Note that the generator of ξ (t) is Now we fix W and , and show that µ is a σ-finite invariant measure for p W by adapting the proof of [13] to the present time-dependent case.
where n is the inner normal vector, e = |ā n| −1 (ā n) and ∂US ∂ e is the directional derivative. Note that e · n = |ā n| −1 n Tā n > 0, so that e points to the interior of S. As U S (t, x) ≥ 0 for x ∈ S and U S (t, x) = 0 for x ∈ ∂S, we have ∂US ∂ e ≥ 0. Hence, we can continue (2.12) with Taking r → ∞, we have Let ρ n be a smooth function on R d such that ρ n (x) = 1 for |x| ≤ n, ρ n (x) = 0 for |x| ≥ 2n and sup By (2.11) and the condition (BC), we have Then Then v n ∈ C([0, T ]) decreases to 0 as n → ∞. By Dini's theorem, v n → 0 uniformly for t ∈ [0, T ]. Therefore, u n (t) → 0 as n → ∞ uniformly for t ∈ [0, T ]. Note that u n (t) → u(t). Therefore, u (t) = lim n→∞ u n (t) = 0. Namely, Let F (W ) be a bounded continuous function of W . Then By the Wong-Zakai theorem (cf. [11] or [5], P410, Theorem 7.2), we have ξ (t) → ξ(t) as → 0. Note that |f (x)| ≤ ce −c1|x| for any c 1 > 0. By Lemma 2, apply the dominated convergence theorem to (2.13), we have This implies the conclusion of the proposition.
2) Suppose that σ 1 (x) =σ 1 (x)I, whereσ 1 is a real-valued function bounded away from 0 and I is the identity matrix. If µ(dx) ≡ 1 σ(x) dx satisfiesL * µ = 0, then the conditions of Proposition 3 hold for µ and hence, µ is an invariant measure for the conditional process.

Remark 5
In general, the σ-finite invariant measure is not unique. Suppose that σ 2 = I and b is a constant vector. As being pointed out in [13], µ 1 (x) = 1 and µ 2 (x) = e 2b T x are two solutions toL * µ = 0. For the second condition, we seek for j = 1, 2, · · · , d. The existence of such σ 1 is clear if d > 2 since there are d 2 entries of σ 1 and 2d < d 2 equations.
3 Non-trivial limit when d ≥ 3 In this section, we extend the process X t to the space of infinite measures and consider the long-time behavior of X t in high spatial dimensions. We shall prove that X t has a non-trivial limit in distribution which is, in fact, persistent. The proof is adopted from Wang [10]. Let P W (·) ≡ P (·|W ) be the conditional probability measure. First, we establish the equivalence between the martingale problem (1.1-1.2) and the conditional martingale problem defined below which is more natural and is easier to handle.
Definition 6 A real valued process U t (adapted to σ-field F t ) is a P W -martingale if for any t > s, E(U t |F t ∨ σ(W )) = U s , a.s.
Lemma 7 X t is a solution to the martingale problem (1.1-1.2) if and only if it is a solution to the following conditional martingale problem (CMP): For all φ ∈ C 2 0 (R d ), is a continuous P W -martingale with quadratic variation process Proof: Suppose that X t is a solution to the martingale problem (1.1-1.2). Similar to the martingale representation Theorem 3.3.6 in Kallianpur and Xiong [6] there exist processes W and B such that W is a R d -valued Brownian motion, B is an L 2 (R d )-cylindrical Brownian motion independent of W , and It is then easy to see that X t solves the CMP (3.1-3.2). On the other hand, suppose that X t is a solution to the CMP (3.1-3.2). As N t (φ) is a P Wmartingale, for s < t, we have where F X t is the σ-field generated by X. Hence the quadratic covariation process N (φ), W t = 0. Therefore, is a martingale with quadratic variation process This proves that X t is a solution to the MP (1.1-1.2). Now, we extend the state space of the superprocess to the space of infinite measures. Let φ a (x) = e −a|x| . Define the space of tempered measures of as: Let S i , i = 1, 2, · · · , be bounded disjoint subsets of R d such that R d = ∪ ∞ i=1 S i , and µ i (·) = µ(· ∩ S i ). Let X i be a sequence of M F (R d )-valued processes which are, given W , conditionally independent and for each i, X i t is a solution to the CMP (3.1-3.2) with µ i in place of µ.
For any a > 0, where the last equality follows from Theorem 5.1 in [12]. By Lemma 2, we have Therefore, we can continue (3.3) with Hence, X t is a well-defined M tem (R d )-valued process. It is easy to show that X t solves the CMP (3.1-3.2), and hence, the MP (1.1-1.2). It is clear that (1.3) remains true for µ ∈ M tem (R d ).
The conclusion of the lemma then follows from the facts that p 0 (t, x, Here is our main result. Theorem 10 Suppose that d ≥ 3, (2.5) holds and µ has density which is bounded by c 1 e c2|x| , where c 1 and c 2 are two constants. Then X t converges in distribution to a limit X ∞ as t → ∞. Furthermore, EX ∞ = µ.
Proof: By (1.4), we have whereW t (r) = W (t) − W (t − r) and the stochastic integral above is the usual Itô integral.
Recall that y s is given by (3.3). Since W andW t are both Brownian motions, {y s : 0 ≤ s ≤ t} and {y t−s,t : 0 ≤ s ≤ t} have the same distribution as stochastic processes. Therefore, Note that y s,t and y s are nonnegative (when f ≥ 0), the above expectations are finite. Taking integral on both sides of (3.4) with respect to the measure µ, by (2.5), we have µ, y t = µ, f − t 0 µ, y 2 r dr. Then, as t → ∞, where the second equality follows from Theorem 5.1 in [12], the inequality follows from (3.4) and the last equality from (2.5). By approximation, we can show that (3.11) still hold if f (x) = e −a|x| . Therefore, {X t } is tight in M tem (R d ). Let X ∞ be a limit point. Then, the Laplace transform of X ∞ is given by the limit on the right hand side of (3.10). Therefore, the limit distribution is unique and hence, X t converges to X ∞ in distribution. By Fatou's lemma, we have where the second inequality follows from (3.11). On the other hand, by Jensen's inequality Replace f by f , we have here y r ( f ) is defined as in (3.3) with f replaced by f . We only need to show that By (3.9), we have where the last equality follows from (3.13) and the dominated convergence theorem.
Therefore, by (3.14), we only need to show that On the other hand, where the first inequality follows from Lemma 9 and the second inequality follows from the well-known fact that R d e c2|x| p 0 (t, x, z)dx ≤ ce c2|z| , the finiteness in the last step of (3.17) follows from the the compact support property imposed on f in the condition (BC). This, together with (3.16), implies the almost sure finiteness in (3.15).
4 Long-time local extinction when d ≤ 2 In this section, we prove the long-term local extinction when d ≤ 2. We adapt the proof of Dawson et al [1] to our present setup.
For any bounded Borel set B in R d , we have For > 0, choose K such that where p t (x) is the density of the normal random vector with mean 0 and covariance matrix tI. Let c and τ be such that f ≤ cp τ . For t > 0, set Note that by (3.3), It is well-known that the above inequality yields ≤ µ, f , here |S r | denotes the Lebesgue measure of S r and g(r) = Sr Ey r (x)µ(dx). As Since is arbitrary, the proof of the statement is complete.