Local behavior of local times of super Brownian motion

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 $L_t^x-(1/\pi )\log (1/|x|)$ 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.

Cited by in corpus (1)

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