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The critical barrier for the survival of branching random walk with absorption. (English. French summary) Zbl 1263.60076
Author’s 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, J. D. Biggins et al. [Ann. Appl. Probab. 1, No. 4, 573–581 (1991; Zbl 0749.60076)] 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 \(a<a_{c}\) 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.

60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
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