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 $t\geq 0$. Then for $d\geq2$ and fixed $t>0$, the singular random measure $\xi_t$ can be a.s. approximated by suitably normalized restrictions of Lebesgue measure to the $\varepsilon$-neighborhoods of $\operatorname {supp}\xi_t$. When $d\geq3$, 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 $\xi$.