Asymptotic behavior of survival probability for a branching random walk with a barrier - You Lv (AIMS Mathematics, 2023)
Consider a branching random walk with a mechanism of elimination. We assume that the underlying Galton-Watson process is supercritical, thus the branching random walk has a positive survival probability. A mechanism of elimination, which is called a barrier, is introduced to erase the particles who lie above $ ri+ε i^{α} $ and all their descendants, where $ i $ presents the generation of the particles, $ α > 1/3, ε\in{{\mathbb{R}}} $ and $ r $ is the asymptotic speed of the left-most position of the branching random walk. First we show that the particle system still has a positive survival probability after we introduce the barrier with $ ε > 0. $ Moreover, we show that the decay of the probability is faster than $ e^{-β'ε^{β}} $ as $ ε\downarrow 0 $, where $ β', β $ are two positive constants depending on the branching random walk and $ α $. The result in the present paper extends a conclusion in Gantert et al. (2011) in some extent. Our proof also works for some time-inhomogeneous cases.