Some local approximations of Dawson--Watanabe superprocesses
Abstract
Description
Let $ξ$ be a Dawson--Watanabe superprocess in $\mathbb{R}^d$ such that $ξ_t$ is a.s. locally finite for every $t\geq 0$. Then for $d\geq2$ and fixed $t>0$, the singular random measure $ξ_t$ can be a.s. approximated by suitably normalized restrictions of Lebesgue measure to the $\varepsilon$-neighborhoods of $\operatorname {supp}ξ_t$. When $d\geq3$, the local distributions of $ξ_t$ near a hitting point can be approximated in total variation by those of a stationary and self-similar pseudo-random measure $\tildeξ$. 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 $ξ$.
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)
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)