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

Abstract

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 $X_{t}^{n}$ converges weakly in the Skorohod space $D([0,T];M_{F}(\mathbb{R} ^{d}))$ and the limit has a density $u_{t}(x)$, where $M_{F}(\mathbb{R} ^{d})$ is the space of finite measures on $\mathbb{R} ^{d}$. We also derive a stochastic partial differential equation $u_{t}(x)$ satisfies. By using the techniques of Malliavin calculus, we prove that $u_{t}(x)$ is jointly Hölder continuous in time with exponent $\frac{1} {2}-\epsilon $ and in space with exponent $1-\epsilon $ for any $\epsilon >0$.