Hölder continuity of the solutions to a class of SPDE’s arising from branching particle systems in a random environment

We consider a d-dimensional branching particle system in a random environment. Suppose that the initial measures converge weakly to a measure with bounded density. Under the Mytnik-Sturm branching mechanism, we prove that the corresponding empirical measure X t converges weakly in the Skorohod space D([0, T ];MF (R )) and the limit has a density ut(x), where MF (R ) is the space of finite measures on R. We also derive a stochastic partial differential equation ut(x) satisfies. By using the techniques of Malliavin calculus, we prove that ut(x) is jointly Hölder continuous in time with exponent 1 2 − and in space with exponent 1− for any > 0.


Introduction
Consider a d-dimensional branching particle system in a random environment. For any integer n ≥ 1, the branching events happen at time k n , k = 1, 2, . . . . The dynamics of each particle, labelled by a multi-index α, is described by the stochastic differential equation (SDE): dx α,n t = dB α t + Gaussian random field on R + × R d independent of the family {B α }. The random field W can be regarded as the random environment for the particle system. The existence and uniqueness of the Feller process x α,n that solves the SDE (1.1) will be proved in Section 2. At any branching time each particle dies and it randomly generates offspring. The new particles are born at the death position of their parents, and inherit the branchingdynamics mechanism. The branching mechanism we use in this paper follows the one introduced by Mytnik [23], and studied further by Sturm [30]. Let X n = {X n t , t ≥ 0} denote the empirical measure of the particle system. One of the main results of this work is to prove that the empirical measure-valued processes converge weakly to a process X = {X t , t ≥ 0}, such that for almost every t ≥ 0, X t has a density u t (x) almost surely.
By using the techniques of Malliavin calculus, we also establish the almost surely joint Hölder continuity of u with exponent 1 2 − in time and 1 − in space for any > 0.
To compare our results with the classical ones. Let us recall briefly some existing work in the literature. The one-dimensional model was initially introduced and studied by Wang ( [32,33]). In these papers, he proved that under the classical Dawson-Watanabe branching mechanism, the empirical measure X n converges weakly to a process X = {X t , t ≥ 0}, which is the unique solution to a martingale problem.
For the above one dimensional model Dawson et al. [8] proved that for almost every t > 0, the limit measure-value process X has a density u t (x) a.s. and u is the weak solution to the following stochastic partial differential equation (SPDE): where h 2 is the L 2 -norm of h, and V is a space-time white Gaussian random field on R + × R independent of W . Suppose further that h is in the Sobolev space H 2 2 (R) and the initial measure has a density µ ∈ H 1 2 (R). Then Li et al. [20] proved that u t (x) is almost surely jointly Hölder continuous. By using the techniques of Malliavin calculus, Hu et al. [14] improved their result to obtain the sharp Hölder continuity: they proved that the Hölder exponents are 1 4 − in time and 1 2 − in space, for any > 0. Our paper is concerned with higher dimensions (d > 1). However in this case, the super Brownian motion (a special case when h = 0) does not have a density (see e.g. Corollary 2.4 of Dawson and Hochberg [6]). Thus in higher dimensional case we have to abandon the classical Dawson-Watanabe branching mechanism and adopt the Mytnik-Sturm one. As a consequence, the difficult term u s (x) in the SPDE (1.2) becomes u s (x) (see equation (3.1) in Section 3 for the exact form of the equation).
We follow the approach introduced in [14] to study the Hölder continuity of the conditional density of a particle motion using Malliavin calculus. However, because of the multidimensional setting considered here, new difficulties arise. On one hand, the integration by parts formulas require higher order Malliavin derivatives which make computations more complex. To lower the order of Malliavin differentiability in our framework, we use the combination of Riesz transform and Malliavin calculus, previously studied in depth by Bally and Caramellino [1] (see Appendix A for the density formula that we are using). Another difficulty is the fact that in the one-dimensional case considered in [14], the Malliavin derivative can be expressed explicitly and this type of formula for the Malliavin derivative is no longer available here. We have to use another approach to obtain appropriate sharp estimates. More details are given in Appendix A. This paper is organized as follows. In Section 2 we shall briefly describe the branching mechanism used in this paper. In Section 3 we state the main results obtained in this paper. These include three theorems. The first one (Theorem 3.3) is about the existence and uniqueness of a (linear) stochastic partial differential equation (equation (3.1)), which is proved (Theorem 3.2) to be satisfied by the density of the limiting empirical measure process X n of the particle system (see (2.12)). The core result of this paper is Theorem 3.4 which intends to give sharp Hölder continuity of the solution u t (x) to (3.1). Section 4 presents the proofs for Theorems 3.2 and 3.3. The proof of Theorem 3.4 is the objective of the remaining sections. First, in Section 5, we focus on the one-particle motion with no branching. By using the techniques from Malliavin calculus, we obtain a Gaussian type estimates for the transition probability density of the particle motion conditional on W . This estimate plays a crucial role in the proof of Theorem 3.4. In Section 6, we derive a conditional convolution representation of the weak solution to the SPDE (3.1), which is used to establish the Hölder continuity. In Section 7, we show that the solution u to (3.1) is Hölder continuous.
Lastly, the martingale problem (4.4)-(4.5) is introduced in Section 4 to prove Theorems 3.2 and 3.3. The well-posedness of the martingale problem can be proved under the assumption that the initial measure has a bounded density. We conjecture that it also holds for an arbitrary finite initial measure. We will not pursue this in this paper (see Remark 4.12 (ii)).

Branching particle system
We split this section into two parts. In Section 2.1, we consider a finite branching-free particle system, and prove the existence and uniqueness of this system. In Section 2.2, we give a brief induction to the Mytnik-Sturm branching mechanism.

Finite branching-free particle system
In this section, we will show the existence and uniqueness of the finite branching-free particle system that is determined by (1.1). The one-dimensional analogue is given by Lemma 1.3 of Wang [32].
Fix a time interval [0, T ]. Let W = {W (t, x), (t, x) ∈ [0, T ] × R d } be a d-dimensional space-time white Gaussian random field. For any positive integer n, let {B i } i∈{1,...,n} be a family of independent d-dimensional Brownian motions that is independent of W . Consider an n-particle system, where the motion of each particle is described by the following stochastic differential equation in a random environment W : with initial condition x i 0 ∈ R d for all i = 1, . . . n. In the case n = 1, we omit all upper indexes in equation (2.1) without confusion.
The following hypothesis for h will be used throughout this paper:  (2.2) where h * = (h ji ) 1≤i,j≤d denotes the transpose of h. Then, for any 1 ≤ i, j ≤ d, and x ∈ R d , by Cauchy-Schwarz's inequality, we have We denote by · 2 the Hilbert Schmidt norm for matrices. Then, by Cauchy-Schwarz's inequality again, we have Similarly, we can show that the first, second, and third partial derivatives of ρ are bounded in R d . We make use of the following notations: ∂ k ∂x i1 · · · ∂x i k ρ ij (x) 2 1 2 , for k = 1, 2, 3. Now let us study the SDE's (2.1). These equations are not coupled and we solve them for each i separately. For this reason in the next theorem, which provides the existence and uniqueness of the equation (for each fixed i), we suppress the superscript index i.
Proof. We prove this theorem by Picard iteration. Let An application of the Itô isometry yields that Noticing that ρ ij has bounded first partial derivatives, we for some constant C independent of m. On the other hand, we can show that By iteration, we can conclude that which is summable in m. In other words, for any t ∈ [0, T ], the sequence x (m) t is convergent in L 2 (Ω). Denote by x t the limit of this sequence. We claim that x = {x t , 0 ≤ t ≤ T } is a strong solution to (2.1) (recall we suppress the superscript). It suffices to show that as m → ∞, While equations (2.1) can be solved separately for each fixed i the solutions x 1 , . . . , x n are not independent since all of them depend on the common random environment W . It is easy to see that (x 1 , . . . , x n ) is an nd-dimensional Feller process governed by the generator A (n) f (y 1 , . . . , y n ) = 1 2 (∆ (n) + B (n) )f (y 1 , . . . , y n ), where ∆ (n) is the Laplace operator in R nd , ∂y i k1 ∂y j k2 (y 1 , . . . , y n ) , (2.7) and y k = (y 1 k , . . . , y d k ) ∈ R d for all k = 1, . . . , n. This is similar to (1.19) of Wang [32] for the one-dimensional case.

The Mytnik-Sturm branching mechanism
In this section, we briefly construct the branching particle system. For further study of this branching mechanism, we refer the readers to Mytnik's and Sturm's papers (see [23,30]).
We start this section by introducing some notation. For any integer k ≥ 0, we denote by C k b (R d ) the space of k times continuously differentiable functions on R d which are bounded together with their derivatives up to the order k. Also, H k 2 (R d ) is the Sobolev space of square integrable functions on R d which have square integrable derivatives up to the order k. For any differentiable function φ on R d , we make use of the notation be a set of multi-indexes. In our model I is the index set of all possible particles. In other words, initially there are a finite number of particles and each particle generates at most 2 offspring. For any particle α = (α 0 , α 1 , . . . , α N ) ∈ I, let α − 1 = (α 0 . . . , α N −1 ), α − 2 = (α 0 , . . . , α N −2 ), . . . , α − N = (α 0 ) be the ancestors of α. Then, |α| = N is the number of the ancestors of the particle α. It is easy to see that the ancestors of any particle α are uniquely determined.
Fix a time interval [0, T ]. Let (Ω, F, P ) be a complete probability space, on which {B α t , t ∈ [0, T ]} α∈I are independent d-dimensional standard Brownian motions, and W is a d-dimensional space-time white Gaussian random field on [0, T ] × R d independent of the family {B α }.
For any t ∈ [0, T ], let t n = nt n be the last branching time before t. For any α = (α 0 , α 1 , . . . , α N ), if nt n = nt ≤ N , let α t = (α 0 , . . . , α nt ) be the ancestor of α at time t. Suppose that each particle, which starts from the death place of its parent, moves in R d following the motion described by the SDE (2.9) during its lifetime. Then, the path of any particle α and all its ancestors, denoted by x α,n t , is given by otherwise.
Here x n α0 ∈ R d is the initial position of particle (α 0 ), x αt−1,n t − n := lim s↑tn x αt−1,n s , and ∂ denotes the "cemetery"-state, that can be understood as a point at infinity.
for all x, y ∈ R d . Assume that ξ satisfies the following conditions: (iii) κ is continuous and bounded on R d × R d .
For any n ≥ 1, the random field ξ is used to define the offspring distribution after a scaling 1 √ n . In order to make the offspring distribution a probability measure, we introduce the truncation of the random field ξ, denoted by ξ n , as follows: (2.11) The correlation function of the truncated random field is then given by Let (ξ n i ) i≥0 be independent copies of ξ n . Denote by ξ n+ i and ξ n− i the positive and negative part of ξ n i respectively. Let N α,n ∈ {0, 1, 2} be the offspring number of the particle α at the branching time |α|+1 n . Assume that {N α,n , |α| = i} are conditionally independent given ξ n i and the position of α at its branching time, with a distribution given by For any particle α = (α 0 , . . . , α N ), α is called to be alive at time t, denoted by α ∼ n t, if the following conditions are satisfied: (i) There are exactly N branching before or at t: nt = N .
[Introduction of N α,n allows the particle α produce one more generation, namely, produce new particle (α, N α,n ). However, (α, 0) is considered no longer alive and will not produce offspring any more.] For any n, denote by X n = {X n t , t ∈ [0, T ]} the empirical measurevalued process of the particle system. Then, X n is a discrete measure-valued process, given by where δ x is the Dirac measure at x ∈ R d , and the sum is among all alive particles at time t ∈ [0, T ]. Then, for any φ ∈ C 2 b (R d ), with the notation (2.8), we have

Main results
Let (Ω, F, {F t } t∈[0,T ] , P ) be a complete filtered probability space that satisfies the usual conditions. Suppose that W is a d-dimensional space-time white Gaussian random field on [0, T ] × R d , and V is a one-dimensional Gaussian random field on [0, T ] × R d independent of W , that is time white and spatially colored with correlation κ defined in (2.10): Denote by A * the adjoint of A, where A = A (1) is the generator defined in (2.6).
Consider the following SPDE: where the last two stochastic integrals are Walsh's integral (see e.g. Walsh [31]).
The solution to (3.1) is said to be pathwise unique, if whenever u and u are two solutions to (3.1), then there exists a set G ∈ F of probability one, such that u t (ω) = u t (ω) for all t ∈ [0, T ] and ω ∈ G.
(ii) u is said to be a weak solution to the SPDE (3.1), if there exists a filtered probability space, on which W and V are independent random fields that satisfy the above properties, such that u is a strong solution with this probability space. (2.12). In order to show the convergence of X n in D([0, T ]; M F (R d )), we make use of the following hypotheses on the initial measures X n 0 : iii) X 0 has a bounded density µ.
In Section 4 we prove the following two theorems.
for almost all ω ∈ Ω and every t ∈ [0, T ], as a finite measure on R d , X t (ω) has a density u t (x, ω).  The last main result in this paper is the following theorem concerning the Hölder continuity of the solution to the SPDE (3.1).

Proof of Theorems 3.2 and 3.3
We prove Theorems 3.2 and 3.3 in the following steps: (i) In Section 4.1, we show that {X n } n≥1 is a tight sequence in D([0, T ]; M F (R d )), and the limit of any convergent subsequence in law solves a martingale problem.
(ii) In Section 4.2, we show that any solution to the martingale problem has a density almost surely.

Tightness and martingale problem
Recall the empirical measure-valued process X n = {X n t , t ∈ [0, T ]} given by (2.12). Let A = A (1) be the generator of one particle motion defined in (2.6). For any φ ∈ C 2 b (R d ), similar to equality (49) of Sturm [30], we can decompose X n t as follows: ∇φ(x α,n u ) * dB α u , U n t (φ) = 1 n sn<tn α∼nsn where N 0 denotes the number of initial particles, that is a finite integer. Therefore, by the stochastic Fubini theorem (see, e.g., Lemma 4.1 on page 116 of Ikeda and Watanabe [15]), we can write: As in Sturm [30], consider the natural filtration, generated by the process X n F n t = σ {x α,n , N α,n |α| < nt } ∪ {x α,n s , s ≤ t, |α| = nt } , and a discrete filtration at branching times F n tn = σ F n tn ∪ {x α,n |α| = nt n } = F n (tn+n −1 ) − .
Then, B n t (φ) and U n t (φ) are continuous F n t -martingales, while M b,n t (φ) is a discrete F n tn -martingale.
Proof. (i) By the same argument as that for Lemma 3.1 of Sturm [30], we can show that where the constant C > 0 does not depend on n. Again similarly as Strum did for (58) of [30], we can also deduce the following inequality where C 1 , C 2 are constants independent of n. Notice that sup 0≤t≤T |X n t (1)| ≤ 2 nT N n 0 n , that is bounded for fixed n. Then, it follows from Gronwall's inequality that the sequence E sup 0≤t≤T |X n t (1)| p n≥1 is uniformly bounded in n.
The uniform boundedness of E sup 0≤t≤T |X n t (φ)| p and E sup 0≤t≤T |M b,n t (φ)| p follows immediately.
We estimate U n t (φ) as follows: Thus by (4.2), Burkholder-Davis-Gundy's and Jensen's inequalities, we have that is also uniformly bounded in n.
(ii) Note that {B α } are independent Brownian motions. Then, by Burkholder-Davis- As a consequence of Lemma 4.1, the collection is uniformly integrable.

Definition 4.2.
Let {X α } be a collection of real-valued stochastic processes. A family of stochastic processes {X α } is said to be C-tight, if it is tight, and the limit of any subsequence is continuous.

Lemma 4.3. Assume hypotheses [H0], [H1], [H2] (i) and (ii). For all
Proof. By an argument that used by Sturm in the proof of Lemma 3.6 in [30], we can deduce the C-tightness of M b,n (φ) and Z n (φ).
We prove the tightness of X n t (φ) by checking Aldous's conditions (see e.g. Theorem 4.5.4 of Dawson [5]). By Chebyshev's inequality, for any fixed t ∈ [0, T ], and N > 0, we On the other hand, for any n ≥ 1, we extend X n to the time interval [0, T n + 1 n ] in such a way that X n performs the same diffusion/branching mechanism as before on [T, n ]} the extended process. Then, by Theorem 10.13 of Dynkin [10], we know that X n is a strong Markov process on [0, T n + 1 n ]. Let {τ n } n≥1 be any collection of stopping times bounded by T and let {δ n } n≥1 be any positive sequence that decreases to 0, such that τ n + δ n ≤ T . Then, due to the uniform boundedness of E sup 0≤t≤T |X n t (φ)| p and the strong Markov property of X n , we have as n → 0. Thus both of Aldous's conditions are satisfied, and then it follows that X n t (φ) is tight in D([0, T ], R).
Recall the decomposition formula (4.1): Notice that X n (φ), Z n (φ), M b,n (φ) are tight sequences as proved just above, X n 0 (φ) converges weakly by assumption, and B n t (φ) converges 0 in L 2 (Ω) uniformly for all t ∈ [0, T ] by Lemma 4.1 (ii). As a consequence, U n (φ) is tight in D([0, T ], R). In addition, by Proposition VI.3.26 of Jacod and Shiryaev [16], every limit of a tight sequence of   On the other hand, by the same argument as in Lemma 3.9 of Sturm [30], we can show that any limit of a weakly convergent subsequence X n k in D( To show property (ii), notice that S ⊂ C 2 b (R d ). Then, by Theorem 4.1 of Mitoma [22], is a continuous and square integrable F X t -adapted martingale with quadratic variation: . By taking further subsequences, we can assume, in view of Therefore, by Skorokhod's representation theorem, there exists a probability space Then, it converges to a continuous and square integrable martingale M (φ) = M b (φ) + U (φ) in L 2 ( Ω) with respect to its natural filtration. The next step is to compute the quadratic variation of M (φ). Notice that W and {B α } are independent, then U n and B n are orthogonal. As a consequence, U n k and B n k are also orthogonal. On the other hand, M b,n (φ) is a pure jump martingale, while U n k (φ) and B n k (φ) are continuous martingales. Due to Theorem 43 on page 353 of Dellacherie and Meyer [9], M b,n (φ), B n k (φ) and U n k (φ) are mutually orthogonal. By the same argument as in Lemma 4 On the other hand, by the same argument of Lemma 3.8 of Sturm [30], we have is a continuously square integrable martingale with the quadratic variation given by the expression (4.5).
Finally, by the same argument as in Theorem II in Section 4.2 of Perkins [26], we can

Assume hypotheses [H0] and [H1]
. Let X t be a solution to the martingale problem (4.4)-(4.5). In this section, we show that for almost every t ∈ [0, T ], as an M F (R d )-valued random variable, X t has a density almost surely. For We derive the moment formula E(X ⊗n t (f )) of the process X. In the one-dimensional Dawson-Watanabe branching case, Skoulakis and Adler [27] obtained the formula by computing the limit of particle approximations. An alternative approach by Xiong [34] consists in differentiating a conditional stochastic log-Laplace equation. In the present paper we use the techniques of moment duality to derive the moment formula. It can be also formulated by computing the limit of particle approximations.
For any integers n ≥ 2 and k ≤ n, we make use of the notation is the correlation of the random field ξ given by (2.10), and A (n) is the generator of n-particle motion defined in (2.6). Lemma 4.6. Let X t be a solution to the martingale problem (4.4)-(4.5). Then, for any n ≥ 2 and f ∈ C 2 b (R nd ), the following process Proof. See Lemma 1.3.2 of Xiong [35].
given by where p is the transition density of n-particle-motion.
k , controlled by i.i.d. exponential clocks τ k . In between jumps, the process evolves deterministically by the continuous semigroup T (n) t . Notice that the exponential clock is memoryless, and the semigroup T (n) t is generated by a time homogeneous Markov process. Therefore, Y (n) is also time homogeneous.
Proof. Since T (n) t is the semigroup generated by a Markov process, for any t > 0 and (4.8) Notice that η j is the sum of i.i.d. exponential random variables. Then, we have for some t ∈ (0, t). Therefore, (4.7) follows from (4.8) and (4.9).
is a martingale.
Proof. Let µ (n) be any finite measure on R nd . Then, we have For the first term, we have For the second term, since τ 1 , τ 2 are independent, then for any 0 ≤ s ≤ t, we have Note that by Lemma 4.7, |Y for some t ∈ (0, t). Combining (4.11), (4.12), and (4.14), we have By Proposition 4.1.7 of Ethier and Kurtz [11], the following process: is a martingale. Then, the lemma follows by choosing µ (n) = µ ⊗n .
By Lemma 4.6, 4.8 and Corollary 3.2 of Dawson and Kurtz [7], we have the following moment equality: , has a unique solution.
(ii) Let X = {X t , t ∈ [0, T ]} be a solution to the martingale problem (4.4)-(4.5). Then, Proof. Firstly, we claim that the operator A (n) = 1 2 (∆ + B (n) ) is uniformly parabolic in the sense of Friedman (see Section 1.1 of [12]). Because ∆ is uniformly parabolic, then it suffices to analyse the properties of B (n) . For any k = 1, . . . , n, i = 1, . . . , d, and ξ i k ∈ R, where Y (n) is defined by (4.6). By the same argument as we did in the proof of Lemma 4.7, we can show that for Then, by the dominated convergence theorem, we have Let µ (n) be any finite measure on R nd . Recall that the process defined by (4.15) is a martingale, then the following equality follows from Fubini's theorem: In other words, . This solution is unique by the same argument as in solves equation (4.17). Therefore, (4.18) follows from (4.20) and the moment duality (4.16).
In Lemma 4.9, we derived the moment formula for E X (n) t (f ) in the case when n ≥ 2. If n = 1, the dual process only involves a deterministic evolution driven by the semigroup of one particle motion, which makes things much simpler. We write the formula below and skip the proof. Let p(t, x, y) be the transition density of the one particle motion, then The existence of the density of X t will be derived following Wang's idea (see Theorem 2.1 of [32]). For any > 0, x ∈ R d , let p be the heat kernel on R d , that is Proof. Let Γ(t, (y 1 , y 2 ); r, (z 1 , z 2 )) be the fundamental solution to the PDE (4.17) when n = 2 (see Chapter 1 of Friedman [12] for a detailed account on existence and properties of fundamental solutions to parabolic PDEs). We write y = (y 1 , y 2 ) and z = ( is the unique solution to the PDE (4.17) with initial condition v 0 = f . Thus by Lemma By inequality (6.12) of Friedman [12] on page 24, we know that there exists C Γ , λ > 0, such that for any 0 ≤ r < t ≤ T , Therefore, by the semigroup property of heat kernels and Fubini's theorem, we have Γ(t, y; 0, z)p 1+ 2 (z 1 − z 2 )dzX ⊗2 0 (dy)dt. From (4.24), (4.25) and the fact that X 0 ∈ M F (R d ) has a bounded density µ, it follows that (4.21) is true.
Let M be the function on R 2d , given by By (6.13) of Friedman [12] on page 24, we know that Γ(t, y; r, x) is uniformly continuous in the spatial argument for any fixed r and t such that 0 < r < t < T . As a consequence M is continuous. Therefore, by (4.24) and the continuity of M, the function N on R d given by is integrable and continuous everywhere. It follows that  . Assume that the initial measure X 0 ∈ M F (R d ) has a bounded density µ. Then, for almost every t ∈ (0, T ], X t is absolutely continuous with respect to the Lebesgue measure almost surely. Proof. As proved in Lemma 4.10, for any x ∈ R d and n ↓ 0, the sequence {X t (p x n )} n≥1 is Cauchy in L 2 (Ω × R d × [0, T ]). Then, it converges to some square integrable random field. By the same argument as in Theorem 2.1 of Wang [32], we can show that the limit random field is the density of X t almost surely.

Remark 4.12.
(i) The assumption in Proposition 4.11, that the initial measure has a bounded density, cannot be removed. Actually, if we choose X 0 = δ 0 , the Dirac delta mass at 0, then . This is another difference from the one dimensional situation, in which case X 0 (1) < ∞ is enough to imply the existence of the density (see Theorem 2.1 Wang [32] for the Dawson-Watanabe branching model).
(ii) The method of duality is conventionally used to prove the well-posedness of martingale problems arisen from branching mechanisms. In the one-dimensional Dawson-Watanabe model, Wang proved the well-posedness by solving a moment problem (see Section 4 of [33]). This requires a moment bound of the form ∞ n=1 r n E(|X t (1)| n )/n! < ∞ for some positive r. However, this method does not work in our model and here is the explanation. In the next section, we will prove that the density u is a solution to equation (3.1) and when h ≡ 0, we have that E( u t , 1 n ) behaves like c 1 e c2n 1+ for some > 0 (see e.g. Theorem 4.4 of Chen et al. [3] and Theorem 4.3 of Hu et al. [13] for some sharp bounds of similar SPDE's).
Therefore, the condition ∞ n=1 r n E(| u t , 1 | n )/n! < ∞ for some positive r cannot be satisfied in our model. In the next section, we prove the well-posedness of the martingale problem (4.4)-(4.5) by the Yamada-Watanabe argument assuming the existence of the density. Without the existence of the density, we are currently not able to use the moment duality to show the well-posedness of the martingale problem. We are not pursue this in the current paper.

Proof of Theorems 3.2 and 3.3
The proof of Theorems 3.2 and 3.3 is based on the equivalence of the martingale problem (4.4)-(4.5) and the SPDE (3.1).
The equivalence between martingale problems and SDE's in finite dimensions was observed in the 1970s (see Stroock and Varachan [29]). An alternative proof given by Kurtz [19] consists of the "Markov mapping theorem". In a recent paper [2] Biswas et al. generalized this result to the infinite dimensional cases with one noise following Kurtz's idea. Here in the present paper, we establish a similar result with two noises by using the martingale representation theorem.
Then, B 1 and B 2 are H 1 -and H 2 -cylindrical Brownian motion respectively, and they are independent. As a consequence, we have is a S -valued Wiener process with covariance for any φ, ϕ ∈ S . Therefore, by (4.27) and the equivalence of stochastic integrals with respect to Hilbert space valued Brownian motion and Walsh's integrals (see e.g. Proposition 2.6 of Dalang and Quer-Sardanyons [4] for spatial homogeneous noises), u is a weak solution to the SPDE (3.1).
Proof of Theorems 3.2 and 3.3. By Propositions 4.5 and 4.13, the SPDE (3.1) has a weak solution, that can be obtained by the branching particle approximation. We do not prove the continuity here, because later in Section 7, we will show that the solution is not only continuous, but also Hölder continuous. The continuity of u yields an improved version of Proposition 4.11. Namely, if X t is a continuous measure-valued process (e.g. the limit of the particle approximation), then it has a density for all t ∈ [0, T ] almost surely.
In the next step, we prove the pathwise uniqueness of equation (3.1). Assume that u and u are two continuous strong solutions to (3.1) with initial condition µ. Let (3.1), with initial condition µ ≡ 0, that is continuous in two parameters. Thus d is also the density of a solution to the martingale problem (4.4)-(4.5), with initial measure X 0 ≡ 0. 1 By the moment duality (4.16), for any is the dual process defined by (4.6) in the case n = 2. Since d is continuous in t, it follows that u = u almost surely. Therefore, by the Yamada-Watanabe argument (see Yamada and Watanabe [36] and Kurtz [18]), we obtain the strong existence and weak uniqueness of equation ( The following corollary is a direct consequence of Theorem 3.3 and Proposition 4.13. Corollary 4.14. Assume Hypotheses [H0], [H1], and assume that X 0 ∈ M F (R d ) has a bounded density. Then, the martingale problem (4.4)-(4.5) is well-posed.

Moment estimates for one-particle motion
In this section, we focus on the one-particle motion without branching. By using the techniques of Malliavin calculus, we will obtain moment estimates for the transition probability density of the particle motion conditional on the environment W . A brief introduction and several theorems on Malliavin calculus are stated in Appendix A. For a detailed account on this topic, we refer the readers to the book of Nualart [24].
For any 0 ≤ r < t ≤ T , we denote by ξ t = ξ r,x t , the path of one-particle motion, with initial position ξ r = x. It satisfies the SDE We will apply the Malliavin calculus on ξ t with respect to the Brownian motion B. Let H = L 2 ([0, T ]; R d ) be the associated Hilbert space. By the Picard iteration scheme (see e.g. Theorem 2.2.1 of Nualart [24]), we can prove that for any t ∈ (r, T ], ξ t ∈ ∩ p≥1 D 3,p (R d ). Particularly, Dξ t satisfies the following system of SDE's D (k) for any θ ∈ [r, t], and D (k) θ ξ i t = 0 for all θ > t.
In order to simplify the expressions, we rewrite the stochastic integrals in (5.2) as integrals with respect to martingales. To this end, let M = {M t , r ≤ t ≤ T } be the d × d matrix-valued process given by Notice that M t is the sum of stochastic integrals, so it is a matrix-valued martingale. The quadratic covariations of {M ij } d i,j=1 are bounded and deterministic: Lemma 5.1. For any 0 ≤ r < t ≤ T , x ∈ R d , let γ t = γ ξt be the Malliavin matrix of ξ t = ξ r,x t , then γ t is nondegenerate almost surely.
Proof. We prove the lemma following Stroock's idea (see Chapter 8 of Stroock [28]). Let λ θ (t) be the d × d symmetric random matrix given by Then, the Malliavin matrix of ξ t is the integral of λ θ (t):
Denote by · 2 the Hilbert-Schmidt norm of matrices. By Jensen's inequality (see Lemma 8.14 of Stroock [28]), the following inequality holds almost surely It is easy to show that sup θ∈[r,t] β θ (t) 2 2p < ∞ for all p ≥ 1. Therefore, the right-hand side of (5.8) is finite a.s., and thus γ t is nondegenerate almost surely.
We denote by σ t = γ −1 t the inverse of the Malliavin matrix of ξ t . In the following lemma, we obtain some moment estimates for the derivatives of ξ t and σ t . Before estimates, we introduce the following generalized Cauchy-Schwarz's inequality.
This completes the proof of (5.10).
Proof of (5.11). In order to prove (5.11), we rewrite the SDE (5.6) in the following way: Similarly as we did in step (i), by Burkholder-Davis-Gundy's, and Minkowski's inequalities, we can show that the martingale terms satisfies the following inequality ds.    Proof of (5.12). By integrating equation (   To estimate the second and the fourth term, notice that by (5.10), we have max 1≤i,j≤d It follows that Notice that γ t σ t = I, a.s., as a consequence, D (γ t σ t ) = DI ≡ 0. That implies Proof of (5.13). Fix 0 ≤ r < t ≤ T . For any θ 1 , θ 2 ∈ [r, t], let θ = θ 1 ∨ θ 2 . Taking the Malliavin derivative on both sides of (5.4), we have the following SDE: θ2 ξ j3 s W j1 (ds, dy).
In the next lemma, we derive estimates for the moments of increments of the derivatives of ξ t and σ t .
We define the following functionals of ξ t Note that ρ(0) is a symmetric nonnegative definite matrix. As a consequence, I + ρ(0) is strictly positive definite, and thus nondegenerate. Therefore, we can find a nondegenerate matrix M , such that M * (I + ρ(0))M = I. Let η = M ξ, then η = {η t , t ∈ [0, T ]} is a martingale with quadratic covariation By Levy's martingale characterization, η is a d-dimensional Brownian motion. Then, ξ = M −1 η is a Gaussian process, with covariance matrix (5.62).
Since for any t > r, Σ t := Σ t,t = (t − r)(I + ρ(0)) is symmetric and positive definite, the probability density of the Gaussian random vector ξ t is given by Recall that ρ(0) is symmetric and nonnegative definite. Then it has eigenvalues λ 1 ≥ λ 2 ≥ · · · ≥ λ d ≥ 0. Let λ be the diagonal matrix with diagonal elements λ 1 , . . . , λ d . There is an orthogonal matrix U , such that ρ(0) = U * λU . Let k be defined in (5.64). It follows Thus for any nonzero x ∈ R d , we have because (I + λ) −1 − 2kI is a nonnegative diagonal matrix. Thus for any x, y ∈ R d , t > r, On the other hand, we have Denote by P W , E W , and · W p the probability, expectation and L p -norm conditional on W . The following two propositions are estimates for the conditional distribution of ξ.
Then, for any p 1 , p 2 ≥ 1 and y ∈ R d , there exists C > 0, depending on p 1 , p 2 , c, h 2 , and d, such that where k is defined in (5.64) and p = p 1 ∨ p 2 .
Proof. Let p = p 1 ∨ p 2 . Then, by Jensen's inequality, we have We consider two different cases.
and equivalently {|ξ t − y| < ρ} ⊂ {|ξ t − x| ≥ |x−y| 2 }. Then, by Lemma 5.7, we have is the volume of the unit sphere in R d . (ii) On the other hand, suppose that 2ρ > |x − y|. Then |x − y| ≤ 2ρ ≤ 2c √ t − r. Thus by Lemma 5.7 again, we have Denote by p W (r, x; t, y) the transition probability density of ξ conditional on W . In other words, p W (r, x; t, y) is the conditional probability density of ξ t = ξ r,x t . The existence of p W (r, x; t, y) is guaranteed by Theorem A.3. By applying Theorem A.4, we can further obtain the following estimate: Proposition 5.9. For any 0 ≤ r < t ≤ T , p ≥ 1, and y ∈ R d , there exist C > 0, depending on T , d, h 3,2 , p, and q, such that where k is defined in (5.64).

A conditional convolution representation
In this section, we follow the idea of Li et al. (see Section 3 of [20]) to obtain a conditional convolution formulation of the SPDE (3.1). Consider the following SPDE: where W and V are the same random fields as in (3.1), p W is the transition density of ξ t given by (5.1) conditional on W .
In order to define the stochastic integral on the right-hand side of (6.1), we introduce the following filtrations. First, for any t ∈ [0, T ], we set The stochastic integral in (6.1) is defined for all F t -adapted processes. But later we will see that the solution u, as a limit of Picard iteration, is in fact adapted to a smaller filtration defined as follows: for any t ∈ [0, T ], solution to the SPDE (6.1), if the following properties are satisfied: (ii) u is square integrable in the following sense: Then, for any p ≥ 1, the following inequality holds: Proof. We prove the lemma by the Picard iteration. Let u 0 (t, x) ≡ µ(x) and let for all n ≥ 1 and 0 ≤ t ≤ T . Since W and V are independent, then V is a martingale with respect to the filtration {F t } t∈[0,T ] . Notice that for any r ∈ [0, T ], F t includes all the information of W , and p W depends only on W . Then, p W (r, z; t, x) is F r -measurable, and by induction u n−1 (r, z) is F r -measurable for all [r, t] ⊂ [0, T ] and x, z ∈ R d . Thus the stochastic integral is well-defined, and u n is an F t -adapted random field. In addition, we know that p W (r, z; t, x) is G t -measurable, and by induction we can assume that u n−1 (t) is G t -measurable as well. Thus the stochastic integral in (6.6) is G t -measurable. Therefore, the limit of u n (t, x) in L 2 (Ω), if exists, is also G t -measurable. Let d n (t, x) := u n+1 (t, x) − u n (t, x). Then We aim to prove the existence and convergence of {u n } n≥1 in L 2p (Ω; L 2 (R d )) by showing that d * n (t) is summable in n. Then, we will show that the limit is a solution to (6.1). By the definition of u n (t), Burkholder-Davis-Gundy, Minkowski's and Cauchy-Schwarz's inequalities, we have By the Markov property, p W (r, z; t, x) depends only on {W (s, z)−W (r, z), s ∈ (r, t], z ∈ R d }. On the other hand, d n−1 (r, z) depends on V and {W (s, z), s ∈ [0, r], z ∈ R d }. Thus, p W (r, z; t, x) and d n−1 (r, z) are independent. That implies Then, by (6.8), (6.9), Young's convolution inequality, Fubini's theorem and Proposition 5.9, we have where C > 0 depends on p, T , d, h, and κ ∞ .
Thus by iteration, we have d * n (t) ≤ C n t 0 rn 0 · · · r2 0 d * 0 (r 1 )dr 1 · · · dr n . (6.11) To estimate d * 0 , we observe that dx. (6.12) By an argument similar to the proof of (6.10), we can show that d * 0 (t) < C. Therefore, we have d * n (t) ≤C t 0 rn 0 · · · r2 0 1dr 1 . . . dr n = C t n n! . (6.13) Notice that d n (t) is summable in n and the corresponding series is bounded on [0,T]. Therefore, for any fixed t ∈ [0, T ], {u n (t, ·)} n≥0 is convergent in L 2p (Ω; L 2 (R d )). Denote by u t (x) the limit of this sequence. We claim that u = {u t (x), t ∈ [0, T ], x ∈ R d } is a strong solution to (6.1). Clearly u satisfies (6.4) and is G t -adapted. Therefore, it suffices to show that as n → ∞, (6.14) in L 2p (Ω) for all t ∈ [0, T ]. Actually, by Burkholder-Davis-Gundy's, Minkowski's, Young's convolution inequalities, and the fact that {p W (r, z; t, x), x, z ∈ R d } and {u n (r, z) − u(r, z), z ∈ R d } are independent, we can write This implies that (6.14) is true. As we discussed before, the limit u(t, x) is G t -measurable, it follows that u(t, x) is a strong solution to (6.1).
In order to show the uniqueness, we assume that By the Ito isometry, Minkowski's and Young's convolution inequalities and the fact that 2 2 dxdr. (6.15) Notice that by definition, for almost every t ∈ [0, T ]. As a consequence of Gronwall's lemma and the fact that d 0 ≡ 0, inequality (6.15) implies d(t, x) ≡ 0, a.s for almost every (t, x) ∈ [0, T ] × R d . It follows that the solution to (6.1) in the sense of Definition 6.1 is unique. In order to obtain the uniform boundedness (6.5), we need to estimate the following expression when applying the Picard iteration: . By a similar argument as we did before, the following inequality can be proved: where C > 0 is independent of n. Then, inequality (6.5) follows immediately. Proof. Let u = {u t (x), t ∈ [0, T ], x ∈ R d } be the unique solution to the SPDE (6.1), and write Z(dt, dx) = u t (x)V (dt, dx) for all t ∈ [0, T ] and x ∈ R d . Then, it suffices to show that u satisfies the following equation: Denote by As u is the strong solution to (6.1), the following equations are satisfied These properties allow us to write

Proof of Theorem 3.4
In this section, we prove Theorem 3.4 by showing the the Hölder continuity of u t (x) in spatial and time variables separately: Then, for any 0 < s < t ≤ T , x, y ∈ R d β ∈ (0, 1) and p > 1, there exists a constant C depending on T , d, h 3,2 , µ ∞ , κ ∞ , p, and β, such that the following inequalities are satisfied: Then, Theorem 3.4 is simply a corollary of Proposition 7.1. In order to prove Proposition 7.1, we need the following Hölder continuity results for the conditional transition density p W (r, z; t, x): x, y ∈ R d , and β ∈ (0, 1). Then, there exists C > 0, depending on T , d, h 3,2 , p and β, such that the following inequalities are satisfied: Before showing the proof, let us firstly derive a variant of the density formula (A.11).
It will be used in the proof of (7.4). Choose φ ∈ C 2 b (R n ), such that 1 B(0,1) ≤ φ ≤ 1 B(0,4) , and its first and second partial derivatives are all bounded by 1. For any x ∈ R d and ρ > 0, we set φ x ρ := φ( ·−x ρ ). Assume that F satisfies all the properties in Theorem A.3. Let Q n be the n-dimensional Poisson kernel (see (A.10)). Then, the density of F can be represented as follows: ) . (7.11) For I 3 , we compute the integral as follows: Thus combining (7.10)-(7.12), we have It is easy to see that inequality (7.13) holds for all x, y ∈ R d .

A Basic introduction on Malliavin calculus
In this section, we present some preliminaries on the Malliavin calculus. We refer the readers to book of Nualart [24] for a detailed account on this topic. where m is any positive integer, 0 ≤ t 1 < · · · < t m ≤ T , and g : R md → R is a smooth function that has all partial derivatives with at most polynomial growth. We make use of the notation x = x k i 1≤i≤m,1≤k≤d for any element x ∈ R md . The basic Hilbert space associated with B is H = L 2 [0, T ]; R d . In the same way, for any n ≥ 1, the iterated derivative D n G of a random variable of the form (A.1) is a random variable with values in H ⊗n = L 2 [0, T ] n ; R d n . For each p ≥ 1, the iterated derivative D n is a closable and unbounded operator on L p (Ω) taking values in L p (Ω; H ⊗n ). For any n ≥ 1, p ≥ 1 and any Hilbert space V , we can introduce the Sobolev space D n,p (V ) of V -valued random variables as the closure of S with respect to the norm By definition, the divergence operator δ is the adjoint operator of D in L 2 (Ω). More precisely, δ is an unbounded operator on L 2 (Ω; H), taking values in L 2 (Ω). We denote by Dom(δ) the domain of δ. Then, for any u = (u 1 , . . . , u d ) ∈ Dom(δ), δ(u) is characterized by the duality relationship: for all for all G ∈ D 1,2 = D 1,2 (R).

(A.2)
Let F be an n-dimensional random vector, with components F i ∈ D 1,1 , 1 ≤ i ≤ n. We associate to F an n × n random symmetric nonnegative definite matrix, called the Malliavin matrix of F , denoted by γ F . The entries of γ F are defined by Suppose that F ∈ ∩ p≥1 D 2,p (R n ), and its Malliavin matrix γ F is invertible. Denote by σ F the inverse of γ F . Assume that σ ij F ∈ ∩ p≥1 D 1,p for all 1 ≤ i, j ≤ n. Let G ∈ ∩ p≥1 D 1,2 . Then Gσ ij F DF k ∈ Dom(δ) for all 1 ≤ i, j, k ≤ n. For such F and G, we define If furthermore H (i) (F, G) ∈ ∩ p≥1 D 1,p for all 1 ≤ i ≤ n, then we define The following lemma is a Wiener functional version of Lemma 9 of Bally and Caramellino [1].
The next theorem is a density formula using the Riesz transformation. The formula was first introduced by Malliavin and Thalmaier (see Theorem Section 4.23 of [21]), then further studied by Bally and Caramenillo [1].
For any integer n ≥ 2, let Q n be the n-dimensional Poisson kernel. That is, The theorem below is the density formula for a class of differentiable random variables.
The result can be generalized to the case without the assumption p F ∞ < ∞ by the same argument as in Theorem 5 of [1].