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

Bruno Jaffuel

Annales de l'I.H.P. Probabilités et statistiques (2012)

  • Volume: 48, Issue: 4, page 989-1009
  • ISSN: 0246-0203

Abstract

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We study a branching random walk on 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. [Ann. Appl. Probab.1(1991) 573–581] 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 th generation and find an explicit critical value a c such that the process dies when a l t ; a c and survives when a g t ; a c . We also obtain the rate of extinction when a l t ; a c and a lower bound for the population when it survives.

How to cite

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Jaffuel, Bruno. "The critical barrier for the survival of branching random walk with absorption." Annales de l'I.H.P. Probabilités et statistiques 48.4 (2012): 989-1009. <http://eudml.org/doc/272080>.

@article{Jaffuel2012,
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. [Ann. Appl. Probab.1(1991) 573–581] 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 $an^\{1/3\}$ to the position of the barrier for the $n$th generation and find an explicit critical value $a_\{c\}$ such that the process dies when $a&lt;a_\{c\}$ and survives when $a&gt;a_\{c\}$. We also obtain the rate of extinction when $a&lt;a_\{c\}$ and a lower bound for the population when it survives.},
author = {Jaffuel, Bruno},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {branching random walk; survival probability; survival probability absorbing barrier},
language = {eng},
number = {4},
pages = {989-1009},
publisher = {Gauthier-Villars},
title = {The critical barrier for the survival of branching random walk with absorption},
url = {http://eudml.org/doc/272080},
volume = {48},
year = {2012},
}

TY - JOUR
AU - Jaffuel, Bruno
TI - The critical barrier for the survival of branching random walk with absorption
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2012
PB - Gauthier-Villars
VL - 48
IS - 4
SP - 989
EP - 1009
AB - 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. [Ann. Appl. Probab.1(1991) 573–581] 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 $an^{1/3}$ to the position of the barrier for the $n$th generation and find an explicit critical value $a_{c}$ such that the process dies when $a&lt;a_{c}$ and survives when $a&gt;a_{c}$. We also obtain the rate of extinction when $a&lt;a_{c}$ and a lower bound for the population when it survives.
LA - eng
KW - branching random walk; survival probability; survival probability absorbing barrier
UR - http://eudml.org/doc/272080
ER -

References

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  11. [11] J. W. Harris and S. C. Harris. Survival probabilities for branching Brownian motion with absorption. Electron. Commun. Probab.12 (2007) 81–92. Zbl1132.60059MR2300218
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