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January 2013 Local conditioning in Dawson–Watanabe superprocesses
Olav Kallenberg
Ann. Probab. 41(1): 385-443 (January 2013). DOI: 10.1214/11-AOP702


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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Olav Kallenberg. "Local conditioning in Dawson–Watanabe superprocesses." Ann. Probab. 41 (1) 385 - 443, January 2013.


Published: January 2013
First available in Project Euclid: 23 January 2013

zbMATH: 1270.60058
MathSciNet: MR3059202
Digital Object Identifier: 10.1214/11-AOP702

Primary: 60G57 , 60J60 , 60J80

Keywords: Brownian snake , cluster representation , Historical process , local and global approximation , Measure-valued branching diffusions , moment measures and Palm distributions

Rights: Copyright © 2013 Institute of Mathematical Statistics

Vol.41 • No. 1 • January 2013
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