Local conditioning in Dawson-Watanabe superprocesses

Abstract

Consider a locally finite Dawson-Watanabe superprocess $\xi=(\xi_t)$ in $\mathsf{R}^d$ with $d\geq2$. 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 $\tilde{\eta}$ 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 $\varepsilon$-neighborhoods of some points $x_1,\ldots,x_n\in \mathsf{R}^d$. In the limit as $\varepsilon\to0$, the restrictions to those sets are conditionally independent and given by the pseudo-random measures $\tilde{\xi}$ or $\tilde{\eta}$, whereas the contribution to the exterior is given by the Palm distribution of $\xi_t$ at $x_1,\ldots,x_n$. Our proofs are based on the Cox cluster representations of the historical process and involve some delicate estimates of moment densities.