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Persistence and Equilibria of Branching Populations with Exponential Intensity

Part of: Markov processes Stochastic processes

Published online by Cambridge University Press:  04 February 2016

Zakhar Kabluchko
Zakhar Kabluchko*
Affiliation:
Ulm University
*
∗ Postal address: Institute of Stochastics, Ulm University, Helmholtzstrasse 18, 89069 Ulm, Germany. Email address: [email protected]
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Abstract

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We consider a system of independent branching random walks on R which start from a Poisson point process with intensity of the form eλ(du) = e-λudu, where λ ∈ R is chosen in such a way that the overall intensity of particles is preserved. Denote by χ the cluster distribution, and let φ be the log-Laplace transform of the intensity of χ. If λφ'(λ) > 0, we show that the system is persistent, meaning that the point process formed by the particles in the nth generation converges as n → ∞ to a non-trivial point process Πeλχ with intensity eλ. If λφ'(λ) < 0 then the branching population suffers local extinction, meaning that the limiting point process is empty. We characterize point processes on R which are cluster invariant with respect to the cluster distribution χ as mixtures of the point processes Πceλχ over c > 0 and λ ∈ Kst, where Kst = {λ ∈ R: φ(λ) = 0, λφ'(λ) > 0}.

Keywords

Branching random walkPoisson point processexponential intensitypersistencelocal extinctionequilibrium statecluster invariant point process

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60G55: Point processes
Type
Research Article
Information
Journal of Applied Probability , Volume 49 , Issue 1 , March 2012 , pp. 226 - 244
Copyright
© Applied Probability Trust 

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