Surviving particles for subcritical branching processes in random environment

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:119

Article

Bansaye, Vincent^1,2 

[1] Univ Paris 06, Lab Probabil & Modeles Aleatoires, F-75013 Paris, France

[2] CNRS, UMR 7599, F-75013 Paris, France

关键词: Branching process in random environment (BPRE);    Yaglom distribution;    Q-process;    Random walk with negative drift;   

DOI  :  10.1016/j.spa.2008.12.005

来源: Elsevier

【 摘 要 】

The asymptotic behavior of a subcritical Branching Process in Random Environment (BPRE) starting with several particles depends on whether the BPRE is strongly subcritical (SS), intermediate subcritical (IS) or weakly subcritical (WS). In the (SS + IS) case, the asymptotic probability of survival is proportional to the initial number of particles, and conditionally on the survival of the population, only one initial particle survives a.s. These two properties do not hold in the (WS) case and different asymptotics are established, which require new results on random walks with negative drift. We provide an interpretation of these results by characterizing the sequence of environments selected when we condition on the survival of particles. This also raises the problem of the dependence of the Yaglom quasistationary distributions on the initial number of particles and the asymptotic behavior of the Q-process associated with a subcritical BPRE. (C) 2008 Elsevier B.V. All rights reserved.

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