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

Bruno Jaffuel (1)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

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 $a > a_c$.

Mots clés: branching random walk; barrier; survival probability

Domaines: Probabilités [math.PR]

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