Some local approximations of Dawson--Watanabe superprocesses

Abstract

Let $\xi$ be a Dawson--Watanabe superprocess in ${\mathbb{R}}^d$ such that ${\xi }_t$ is a.s. locally finite for every $t\ge 0$. Then for $d\ge 2$ and fixed $t>0$, the singular random measure ${\xi }_t$ can be a.s. approximated by suitably normalized restrictions of Lebesgue measure to the $\epsilon$-neighborhoods of $\mathrm{supp}{\xi }_t$. When $d\ge 3$, the local distributions of ${\xi }_t$ near a hitting point can be approximated in total variation by those of a stationary and self-similar pseudo-random measure $\tilde{\xi }$. By contrast, the corresponding distributions for $d=2$ are locally invariant. Further results include improvements of some classical extinction criteria and some limiting properties of hitting probabilities. Our main proofs are based on a detailed analysis of the historical structure of $\xi$.

Published in at http://dx.doi.org/10.1214/07-AOP386 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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