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November 2008 Some local approximations of Dawson–Watanabe superprocesses
Olav Kallenberg
Ann. Probab. 36(6): 2176-2214 (November 2008). DOI: 10.1214/07-AOP386


Let ξ be a Dawson–Watanabe superprocess in ℝd such that ξt is a.s. locally finite for every t≥0. Then for d≥2 and fixed t>0, the singular random measure ξt can be a.s. approximated by suitably normalized restrictions of Lebesgue measure to the ɛ-neighborhoods of supp ξt. When d≥3, the local distributions of ξt near a hitting point can be approximated in total variation by those of a stationary and self-similar pseudo-random measure ξ̃. 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.


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Olav Kallenberg. "Some local approximations of Dawson–Watanabe superprocesses." Ann. Probab. 36 (6) 2176 - 2214, November 2008.


Published: November 2008
First available in Project Euclid: 19 December 2008

zbMATH: 1167.60010
MathSciNet: MR2478680
Digital Object Identifier: 10.1214/07-AOP386

Primary: 60G57 , 60J60 , 60J80

Keywords: historical clusters , hitting probabilities , local distributions , Local extinction , Measure-valued branching diffusions , moment densities , neighborhood measures , Palm distributions , self-similarity , Super-Brownian motion

Rights: Copyright © 2008 Institute of Mathematical Statistics


Vol.36 • No. 6 • November 2008
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