Criterion on the limits of superprocesses
type: Item
instance of: scholarly article
title: Criterion on the limits of superprocesses
zbMATH Open document ID: 0920.60065
DOI: 10.1007/BF02871668
published in: Science in China. Series A
publication date: 1998-11-25T00:00:00Z
review text: The author considers superprocesses with an underlying process which is a diffusion on \(\mathbb{R}^d\) satisfying certain natural conditions, and a branching function of the form \[ \Phi(z)= bz+cz^2+ \int^\infty_0 (e^{-zu}-1+zu) n(du), \] where \(c\geq 0\), \(b\in\mathbb{R}\) and \(n\) is a measure on \((0,\infty)\) such that \(\int^\infty_0 u\wedge u^2n(du)<\infty\). The objective of the paper is to study the behavior of such superdiffusions as time \(t\to\infty\), mainly the usual problems of extinction, persistence and equilibrium states in the case of infinite measure values. These problems have been studied extensively for branching functions of the form \(\Phi(z)=c z^{1+\beta}\), \(c>0\), \(0<\beta\leq 1\) [see, e.g. the reviewer and \textit{A. Wakolbinger}, Ann. Probab. 19, No. 1, 266-288 (1991; Zbl 0732.60093)], \textit{M. Bramson}, \textit{J. T. Cox} and \textit{A. Greven}, ibid. 25, No. 1, 56-70 (1997; Zbl 0882.60091) and \textit{A. Klenke}, ibid. 26, 1057-1087 (1998)] using several methods (e.g. Campbell measures and backward trees). The author employs a different method. Assuming that the Lebesgue measure \(m\) is invariant for the underlying diffusion, and starting the superdiffusion from \(m\), it is shown that the superdiffusion has a nontrivial equilibrium state (resp. has only the zero equilibrium state) if \(b<0\) (resp. \(b>0)\). If \(b=0\) and in addition \(\int^\infty_0u^2n(du)<\infty\), then extinction (resp. persistence) holds when \(d<3\) (resp. \(d\geq 3)\).
Mathematics Subject Classification ID: 60J80
Mathematics Subject Classification ID: 60G57
zbMATH DE Number: 1207972
zbMATH Keywords: superprocess
zbMATH Keywords: branching diffusion
zbMATH Keywords: extinction
zbMATH Keywords: equilibrium state
author: Yongjin Wang
reviewed by: Luis G. Gorostiza
MaRDI profile type: Publication
cites work: Q3142396
cites work: Q4866046
cites work: The critical measure diffusion process
cites work: Historical processes
cites work: Q3481021
cites work: Persistence criteria for a class of critical branching particle systems in continuous time
cites work: Q4086524
cites work: Q3862204
cites work: Branching particle systems and superprocesses
cites work: Construction and regularity of measure-valued Markov branching processes
cites work: A weighted occupation time for a class of measure-valued branching processes
cites work: A proof of the persistence criterion for a class of superprocesses
full work available at URL: https://doi.org/10.1007/bf02871668
OpenAlex ID: W2067953883
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description: scientific article; zbMATH DE number 1207972