Occupation times for superprocesses in random environments

Let $X=(X_t, t\geq 0)$ be a superprocess in a random environment governed by a Gaussian noise $W=\{W(t, x),t\geq 0,x\in\mathbb{R}^d\}$ 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 $d\leq 3$, whereas it is singular with respect to Lebesgue measure in $d\geq 4$. Regarding the absolutely continuous case in $d\leq 3$, we further prove that the associated density function is jointly H\"older continuous based on the Tanaka formula and moment formulas, and derive the H\"older exponents with respect to the spatial variable $x$ and the time variable $t$.