On the boundary local time measure of super-Brownian motion

TL;DR

This paper constructs a boundary local time measure for super-Brownian motion in 2 and 3 dimensions, confirming a conjecture, and relates it to the boundary of the range, providing explicit moment measures and refining previous bounds.

Contribution

It introduces a new boundary local time measure for super-Brownian motion, confirming a conjecture, and links it to the boundary of the process's range with explicit moment calculations.

Findings

Support of the boundary local time equals the topological boundary of the range.

Derived explicit first and second moment measures using negative dimensional Bessel processes.

Refined $L^2$ asymptotics and tail bounds for local times.

Abstract

If $L^x$ is the total occupation local time of $d$-dimensional super-Brownian motion, $X$, for $d=2$ and $d=3$, we construct a random measure $\mathcal{L}$, called the boundary local time measure, as a rescaling of $L^x e^{-\lambda L^x} dx$ as $\lambda\to \infty$, thus confirming a conjecture of \cite{MP17} and further show that the support of $\mathcal{L}$ equals the topological boundary of the range of $X$, $\partial\mathcal{R}$. This latter result uses a second construction of a boundary local time $\widetilde{\mathcal{L}}$ given in terms of exit measures and we prove that $\widetilde{\mathcal{L}}=c\mathcal{L}$ a.s. for some constant $c>0$. We derive reasonably explicit first and second moment measures for $\mathcal{L}$ in terms of negative dimensional Bessel processes and use it with the energy method to give a more direct proof of the lower bound of the Hausdorff dimension of $\partial\mathcal{R}$ in \cite{HMP18}. The construction requires a refinement of the $L^2$ upper bounds in \cite{MP17} and \cite{HMP18} to exact $L^2$ asymptotics. The methods also refine the left tail bounds for $L^x$ in \cite{MP17} to exact asymptotics. We conjecture that the Minkowski content of $\partial\mathcal{R}$ is equal to the total mass of the boundary local time $\mathcal{L}$ up to some constant.