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Olav Kallenberg

Publications and source records attributed to Olav Kallenberg.

2 recordsLinked to original sources
arXiv2013

Local conditioning in Dawson-Watanabe superprocesses

Consider a locally finite Dawson-Watanabe superprocess $ξ=(ξ_t)$ in $\mathsf{R}^d$ with $d\geq2$. 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 $ξ_t$ by a stationary cluster $\tildeη$ with nice continuity and scaling properties. This all leads up to an asymptotic description of the conditional distribution of $ξ_t$ for a fixed $t>0$, given that $ξ_t$ 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 $\tildeξ$ or $\tildeη$, whereas the contribution to the exterior is given by the Palm distribution of $ξ_t$ 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.

math.PR↗
arXiv2009

Some local approximations of Dawson--Watanabe superprocesses

Let $ξ$ be a Dawson--Watanabe superprocess in $\mathbb{R}^d$ such that $ξ_t$ is a.s. locally finite for every $t\geq 0$. Then for $d\geq2$ and fixed $t>0$, the singular random measure $ξ_t$ can be a.s. approximated by suitably normalized restrictions of Lebesgue measure to the $\varepsilon$-neighborhoods of $\operatorname {supp}ξ_t$. When $d\geq3$, the local distributions of $ξ_t$ near a hitting point can be approximated in total variation by those of a stationary and self-similar pseudo-random measure $\tildeξ$. By contrast, the corresponding distributions for $d=2$ are locally invariant. Further results include improvements of some classical extinction criteria and some limiting properties of hitting probabilities. Our main proofs are based on a detailed analysis of the historical structure of $ξ$.

math.PR↗