Absolute continuity of the Super-Brownian motion with infinite mean

TL;DR

This paper proves that super-Brownian motion with infinite mean is absolutely continuous at any fixed time, and establishes uniqueness and continuous dependence of solutions for the associated log-Laplace equation.

Contribution

It demonstrates absolute continuity of super-Brownian motion with infinite mean and proves uniqueness and stability of solutions to the log-Laplace equation.

Findings

Super-Brownian motion is absolutely continuous for all dimensions and parameters.

Unique, continuous solutions exist for the log-Laplace equation with finite, non-zero initial data.

Solutions depend continuously on initial conditions.

Abstract

In this work we prove that for any dimension $d\geq 1$ and any $\gamma \in (0,1)$ super-Brownian motion corresponding to the log-Laplace equation \begin{equation*} \begin{split} \frac{\partial v(t,x)}{\partial t } & = \frac{1}{2}\bigtriangleup v(t,x) + v^\gamma (t,x) ,\: (t,x) \in \mathbb{R}_+\times \mathbb{R}^d,\\ v(0,x)&= f(x) \end{split} \end{equation*} is absolutely continuous with respect to the Lebesgue measure at any fixed time $t>0$. Our proof is based on properties of solutions of the \LL\ equation. We also prove that when initial datum $v(0,\cdot)$ is a finite, non-zero measure, then the \LL\ equation has a unique, continuous solution. Moreover this solution continuously depends on initial data.