The critical random barrier for the survival of branching random walk with absorption

Abstract : We study a branching random walk on $\mathbb{R}$ with an absorbing barrier. The position of the barrier depends on the generation. In each generation, only the individuals born below the barrier survive and reproduce. Given a reproduction law, Biggins et al. [4] determined whether a linear barrier allows the process to survive. In this paper, we refine their result: in the boundary case in which the speed of the barrier matches the speed of the minimal position of a particle in a given generation, we add a second order term $a n^{1/3}$ to the position of the barrier for the $n^\mathrm{th}$ generation and find an explicit critical value $a_c$ such that the process dies when $a < a_c$ and survives when $a > a_c$. We also obtain the rate of extinction when $a < a_c$ and a lower bound on the surviving population when $a > a_c$.
Type de document :
Pré-publication, Document de travail
2009
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https://hal.archives-ouvertes.fr/hal-00430791
Contributeur : Bruno Jaffuel <>
Soumis le : mardi 10 novembre 2009 - 23:27:09
Dernière modification le : jeudi 20 juillet 2017 - 09:28:10
Document(s) archivé(s) le : jeudi 17 juin 2010 - 17:57:22

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  • HAL Id : hal-00430791, version 1
  • ARXIV : 0911.2227

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Bruno Jaffuel. The critical random barrier for the survival of branching random walk with absorption. 2009. <hal-00430791>

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