Local conditioning in Dawson–Watanabe superprocesses

Consider a locally finite Dawson-Watanabe superprocess xi = (xi(t)) in R-d with d >= 2. Our main results include some recursive formulas for the moment measures of xi, 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 xi(t) by a stationary cluster (eta) over tilde with nice continuity and scaling properties. This all leads up to an asymptotic description of the conditional distribution of xi(t) for a fixed t > 0, given that xi(t) charges the epsilon-neighborhoods of some points x(1), ..., x(n) is an element of R-d. In the limit as epsilon -> 0, the restrictions to those sets are conditionally windependent and given by the pseudo-random measures (xi) over tilde or (eta) over tilde, whereas the contribution to the exterior is given by the Palm distribution of xi(t) at x(1),..., x(n). Our proofs are based on the Cox cluster representations of the historical process and involve some delicate estimates of moment densities.