Local behavior of local times of super Brownian motion

TL;DR

This paper investigates the asymptotic behavior of the local time of super-Brownian motion near zero in dimensions 2 and 3, establishing normal convergence and boundedness results through a refined Tanaka formula.

Contribution

It introduces new asymptotic results for local times of super-Brownian motion and refines Tanaka's formula for this context.

Findings

In 3D, normalized local time converges to a normal distribution.

In 2D, local time remains bounded in L^1 as x approaches zero.

Provides a refined Tanaka formula for super-Brownian local time.

Abstract

For $x\in R^d- \{0\}$, in dimension $d=3$, we study the asymptotic behavior of the local time $L_t^x$ of super-Brownian motion $X$ starting from $\delta_0$ as $x \to 0$. Let $\psi(x)=((1/2\pi^2) \log (1/|x|))^{1/2}$ be a normalization, Theorem 1 implies that $(L_t^x-(1/2\pi|x|))/\psi(x)$ converges in distribution to a standard normal distributed random variable as $x\to 0$. For dimension $d=2$, Theorem 2 implies that $L^x_t-(1/\pi)\log(1/|x|)$ is $L^1$ bounded as $x\to 0$. To do this, we prove a Tanaka formula for the local time which refines a result in Barlow, Evans and Perkins.