Occupation times for superprocesses in random environments

TL;DR

This paper investigates the occupation times of superprocesses in Gaussian random environments, revealing dimension-dependent absolute continuity and regularity properties of the occupation density.

Contribution

It establishes the absolute continuity of occupation times in dimensions up to 3 and proves joint Hölder continuity of the density, with explicit exponents, in these cases.

Findings

Occupation time is absolutely continuous in dimensions d ≤ 3.

Occupation time is singular in dimensions d ≥ 4.

The density function is jointly Hölder continuous in space and time for d ≤ 3.

Abstract

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 $t$.