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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. [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 an1/3 to the position of the barrier for the nth generation and find an explicit critical value ac such that the process dies when a ac. We also obtain the rate of extinction when a < ac and a lower bound for the population when it survives.

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