Local conditioning in Dawson-Watanabe superprocesses

Abstract

Consider a locally finite Dawson-Watanabe superprocess $\xi =({\xi }_t)$ in ${\mathsf{R}}^d$ with $d\ge 2$. 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 $\tilde{\eta }$ 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 $\epsilon$-neighborhoods of some points $x_1,\dots ,x_n\in {\mathsf{R}}^d$. In the limit as $\epsilon \to 0$, 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 $x_1,\dots ,x_n$. Our proofs are based on the Cox cluster representations of the historical process and involve some delicate estimates of moment densities.

Published in at http://dx.doi.org/10.1214/11-AOP702 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (1)

• Some local approximations of Dawson--Watanabe superprocesses