Persistence criteria for a class of critical branching particle systems in continuous time
type: Item
instance of: scholarly article
title: Persistence criteria for a class of critical branching particle systems in continuous time
zbMATH Open document ID: 0732.60093
published in: The Annals of Probability
publication date: 1991-01-01T00:00:00Z
review text: This paper is concerned with the long-time behavior of a branching particle system in \({\mathbb{R}}^ d\) with spatially homogeneous initial Poisson random field. The particles follow \(\alpha\)-symmetric stable processes and the offspring distributions are chosen from a class of distributions in the domain of attraction of a stable law with index \(1+\beta\). The central result of this paper is the determination of a formula (based on Kallenberg's backward technique (1977)) for the Palm distributions of these branching particle systems [the analogous result for superprocesses is given by the reviewer and \textit{E. A. Perkins}, Mem. Am. Math. Soc. 454 (1991)]. The Palm formula is then used to establish that the branching particle system is persistent (i.e. has a stationary distribution preserving the initial intensity) if and only if \(d>\alpha /\beta\).
Mathematics Subject Classification ID: 60J80
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 60G57
Mathematics Subject Classification ID: 60G55
zbMATH DE Number: 4211224
zbMATH Keywords: long-time behavior of a branching particle system
zbMATH Keywords: Palm distributions
zbMATH Keywords: superprocesses
zbMATH Keywords: stationary distribution
author: Luis G. Gorostiza
author: Anton Wakolbinger
reviewed by: Donald A. Dawson
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176990544
OpenAlex ID: W2036857451
DOI: 10.1214/AOP/1176990544
description: scientific article; zbMATH DE number 4211224