Some local approximations of Dawson–Watanabe superprocesses

Abstract

Let ξ be a Dawson–Watanabe superprocess in ℝ^d such that ξ_t is a.s. locally finite for every t≥0. Then for d≥2 and fixed t>0, the singular random measure ξ_t can be a.s. approximated by suitably normalized restrictions of Lebesgue measure to the ɛ-neighborhoods of supp ξ_t. When d≥3, 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 ξ̃. 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.