Absolute continuity of the super-Brownian motion with infinite mean

Abstract

In this work, we prove that for any dimension $\mathit{d}\ge 1$ and any $\mathit{\gamma }\in (0,1)$ super-Brownian motion corresponding to the log-Laplace equation

$$\mathit{v}(\mathit{t},\mathit{x})=({\mathit{S}}_{\mathit{t}}\mathit{f})(\mathit{x})-{\int }_0^{\mathit{t}}({\mathit{S}}_{\mathit{t}-\mathit{s}}{\mathit{v}}^{\mathit{\gamma }}(\mathit{s},·))(\mathit{x})\mathrm{d}\mathit{s},(\mathit{t},\mathit{x})\in {\mathbb{R}}_{+}\times {\mathbb{R}}^{\mathit{d}},$$

is absolutely continuous with respect to Lebesgue measure at any fixed time $\mathit{t}>0$. ${\{{\mathit{S}}_{\mathit{t}}\}}_{\mathit{t}\ge 0}$ denotes a transition semigroup of a standard Brownian motion. Our proof is based on properties of solutions of the log-Laplace equation. We also prove that when initial datum $\mathit{v}(0,·)$ is a finite, non-zero measure, then the log-Laplace equation has a unique, continuous solution. Moreover this solution continuously depends on initial data.