Some local approximations of Dawson–Watanabe superprocesses

Let xi be a Dawson-Watanabe superprocess in R-d such that xi(t) is a.s. locally finite for every t >= 0. Then for d >= 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 supp xi(t). When d >= 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 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.