Electronic Journal of Probability

Critical Multitype Branching Systems: Extinction Results

Peter Kevei and Jose Lopez Mimbela

Full-text: Open access

Abstract

We consider a critical branching particle system in $\mathbb{R}^d$, composed of individuals of a finite number of types $i\in\{1,\ldots,K\}$. Each individual of type i moves independently according to a symmetric $\alpha_i$-stable motion. We assume that the particle lifetimes and offspring distributions are type-dependent. Under the usual independence assumptions in branching systems, we prove extinction theorems in the following cases: (1) all the particle lifetimes have finite mean, or (2) there is a type whose lifetime distribution has heavy tail, and the other lifetimes have finite mean. We get a more complex dynamics by assuming in case (2) that the most mobile particle type corresponds to a finite-mean lifetime: in this case, local extinction of the population is determined by an interaction of the parameters (offspring variability, mobility, longevity) of the long-living type and those of the most mobile type. The proofs are based on a precise analysis of the occupation times of a related Markov renewal process, which is of independent interest.

Article information

Source
Electron. J. Probab., Volume 16 (2011), paper no. 50, 1356-1380.

Dates
Accepted: 9 August 2011
First available in Project Euclid: 1 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464820219

Digital Object Identifier
doi:10.1214/EJP.v16-908

Mathematical Reviews number (MathSciNet)
MR2827463

Zentralblatt MATH identifier
1245.60082

Subjects
Primary: MSC 60J80
Secondary: MSC 60K15

Keywords
Critical branching particle system Extinction Markov renewal process

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Kevei, Peter; Lopez Mimbela, Jose. Critical Multitype Branching Systems: Extinction Results. Electron. J. Probab. 16 (2011), paper no. 50, 1356--1380. doi:10.1214/EJP.v16-908. https://projecteuclid.org/euclid.ejp/1464820219


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