The critical barrier for the survival of branching random walk with absorption
Jaffuel, Bruno
Annales de l'I.H.P. Probabilités et statistiques, Tome 48 (2012), p. 989-1009 / Harvested from Numdam
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Nous étudions une marche aléatoire branchante sur ℝ 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.

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 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.

Publié le : 2012-01-01
DOI : https://doi.org/10.1214/11-AIHP453
Classification:  60J80 
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     author = {Jaffuel, Bruno},
     title = {The critical barrier for the survival of branching random walk with absorption},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     volume = {48},
     year = {2012},
     pages = {989-1009},
     doi = {10.1214/11-AIHP453},
     mrnumber = {3052402},
     zbl = {1263.60076},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPB_2012__48_4_989_0}
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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, Tome 48 (2012) pp. 989-1009. doi : 10.1214/11-AIHP453. http://gdmltest.u-ga.fr/item/AIHPB_2012__48_4_989_0/

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