Resumen de Local conditioning in Dawson–Watanabe superprocesses

Olav Kallenberg

• Consider a locally finite Dawson–Watanabe superprocess ξ=(ξt) in Rd with d≥2. Our main results include some recursive formulas for the moment measures of ξ, with connections to the uniform Brownian tree, a Brownian snake representation of Palm measures, continuity properties of conditional moment densities, leading by duality to strongly continuous versions of the multivariate Palm distributions, and a local approximation of ξt by a stationary cluster η~ with nice continuity and scaling properties. This all leads up to an asymptotic description of the conditional distribution of ξt for a fixed t>0, given that ξt charges the ε-neighborhoods of some points x1,…,xn∈Rd. In the limit as ε→0, the restrictions to those sets are conditionally independent and given by the pseudo-random measures ξ~ or η~, whereas the contribution to the exterior is given by the Palm distribution of ξt at x1,…,xn. Our proofs are based on the Cox cluster representations of the historical process and involve some delicate estimates of moment densities.