H\"{o}lder continuity of the solutions to a class of SPDEs arising from multidimensional superprocesses in 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 $X_t^n$ 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 H\"{o}lder continuous in time with exponent $\frac{1}{2}-\epsilon$ and in space with exponent $1-\epsilon$ for any $\epsilon>0$.