The Annals of Probability

Local conditioning in Dawson–Watanabe superprocesses

Olav Kallenberg

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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.

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Ann. Probab., Volume 41, Number 1 (2013), 385-443.

First available in Project Euclid: 23 January 2013

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Zentralblatt MATH identifier

Primary: 60G57: Random measures 60J60: Diffusion processes [See also 58J65] 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Measure-valued branching diffusions moment measures and Palm distributions local and global approximation historical process cluster representation Brownian snake


Kallenberg, Olav. Local conditioning in Dawson–Watanabe superprocesses. Ann. Probab. 41 (2013), no. 1, 385--443. doi:10.1214/11-AOP702.

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