Persistence of critical multitype particle and measure branching processes
type: Item
instance of: scholarly article
title: Persistence of critical multitype particle and measure branching processes
zbMATH Open document ID: 0767.60090
DOI: 10.1007/BF01300559
author: Sylvie Rœlly
author: Luis G. Gorostiza
author: Anton Wakolbinger
publication date: 1993-03-22T00:00:00Z
review text: The paper considers multitype branching particle systems with a special branching law which corresponds to mutations of types and monotype branchings, and their corresponding high density limits known as measure branching processes, Dawson-Watanabe (DW) processes, or superprocesses. More precisely, the branching particle system under consideration consists of particles of types \(i=1,\dots,k\); particles of type \(i\) move in \(\mathbb{R}^ d\) for an exponential type-dependent lifetime according to a spherically symmetric stable process with exponent \(\alpha_ i\in(0,2]\), they can eventually mutate into any other type, and they can produce particles of their own type according to a critical branching law which belongs to the domain of attraction of a stable law with exponent \(1+\beta_ i\), \(\beta_ i\in(0,1]\). The question of persistence of the system is investigated by a method due to Kallenberg, the main tool for applying this method being a representation of the Palm distributions of the branching particle system in terms of a ``backward tree'' (Theorem A). The persistence criterion for the multitype branching particle system (Theorem B) says that, if at time 0 the particles are distributed according to a Poisson random measure on \(\mathbb{R}^ d\times\{1,\dots,k\}\) with a product intensity measure \(\lambda\) whose ``position component'' is Lebesgue measure and whose ``type proportions'' are stationary in time with respect to the process of mutations, then the system converges in distribution towards an equilibrium with intensity \(\lambda\) if and only if \(d>\min(\alpha_ 1,\dots,\alpha_ k)/\min(\beta_ 1,\dots,\beta_ k)\). In case of non- persistence the system goes to local extinction in the large time limit, i.e., the number of particles in any bounded subset of \(\mathbb{R}^ d\) tends to 0 in probability. This result means that persistence holds when the mobility of the system is large enough to fill the holes produced by the tendency to extinction of th…
Mathematics Subject Classification ID: 60J80
zbMATH DE Number: 112841
zbMATH Keywords: Palm distribution
zbMATH Keywords: backward tree
zbMATH Keywords: persistence
zbMATH Keywords: multitype branching particle systems
zbMATH Keywords: measure branching processes
zbMATH Keywords: domain of attraction of a stable law
zbMATH Keywords: Poisson random measure
zbMATH Keywords: Cox representation
reviewed by: Luis G. Gorostiza
MaRDI profile type: Publication
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published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 112841