Occupation times for superprocesses in random environments

Abstract

Let $X=(X_t,t\ge 0)$ be a superprocess in a random environment governed by a Gaussian noise $W=\{W(t,x),t\ge 0,x\in {\mathbb{R}}^d\}$ white in time and colored in space with correlation kernel $g$. We consider the occupation time process of the model starting from a finite measure. It is shown that the occupation time process of $X$ is absolutely continuous with respect to Lebesgue measure in $d\le 3$, whereas it is singular with respect to Lebesgue measure in $d\ge 4$. Regarding the absolutely continuous case in $d\le 3$, we further prove that the associated density function is jointly Hölder continuous based on the Tanaka formula and moment formulas, and derive the Hölder exponents with respect to the spatial variable $x$ and the time variable $t$.