The critical barrier for the survival of branching random walk with absorption
Annales de l'I.H.P. Probabilités et statistiques, Volume 48 (2012) no. 4, pp. 989-1009.

We study a branching random walk on 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 nth 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 for the population when it survives.

Nous étudions une marche aléatoire branchante sur avec une barrière absorbante. La position de la barrière dépend de la génération. À chaque génération, seuls les individus nés sous la barrière survivent et se reproduisent. Étant donnée une loi de reproduction, Biggins et al. [Ann. Appl. Probab. 1 (1991) 573-581] ont déterminé, pour une barrière linéaire, si le processus survit ou s’éteint. Dans cet article, nous affinons ce résultat : dans le cas frontière où la vitesse de la barrière correspond à la vitesse de la particule la plus à gauche d’une génération donnée, nous allons à l’ordre suivant en ajoutant un terme an 1/3 à la position de la barrière pour la nième génération et obtenons une valeur critique explicite a c telle que le processus s’éteint quand a<a c et survit quand a>a c . Nous obtenons aussi le taux d’extinction lorsque a<a c et une borne inférieure sur la taille de la population lorsqu’il survit.

DOI: 10.1214/11-AIHP453
Classification: 60J80
Keywords: branching random walk, survival probability
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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, Volume 48 (2012) no. 4, pp. 989-1009. doi : 10.1214/11-AIHP453. http://www.numdam.org/articles/10.1214/11-AIHP453/

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