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

Abstract

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. \cite{BLSW91} 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 $a{a}_c$. We also obtain the rate of extinction when $a<a_c$ and a lower bound on the surviving population when $a>a_c$.

References in corpus (1)

• Quasi-stationary regime of a branching random walk in presence of an absorbing wall

Cited by in corpus (1)

• Weak convergence for the minimal position in a branching random walk: a simple proof