H\"{o}lder continuity of the solutions to a class of SPDEs arising from multidimensional superprocesses in random environment

TL;DR

This paper studies the regularity of solutions to a class of SPDEs derived from multidimensional superprocesses in random environments, proving their joint Hölder continuity using Malliavin calculus.

Contribution

It establishes the Hölder continuity of solutions to SPDEs from superprocesses in random environments, a novel regularity result in this context.

Findings

The empirical measure converges weakly to a density function.

The solution to the SPDE is jointly Hölder continuous in time and space.

The regularity exponents are approximately 1/2 in time and 1 in space.

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 H\"{o}lder continuous in time with exponent $\frac{1}{2}-\epsilon$ and in space with exponent $1-\epsilon$ for any $\epsilon>0$.