Local conditioning in Dawson-Watanabe superprocesses

TL;DR

This paper develops recursive formulas and local approximation techniques for Dawson-Watanabe superprocesses, providing an asymptotic description of their conditional distributions given charge in small neighborhoods, with connections to Brownian trees and Palm measures.

Contribution

It introduces new recursive formulas for moment measures, a Brownian snake representation of Palm measures, and a local approximation of superprocesses, advancing understanding of their conditional distributions.

Findings

Recursive formulas for moment measures of superprocesses

A local approximation of superprocesses by stationary clusters

Asymptotic independence of restrictions conditioned on small neighborhoods

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.