Asymptotic behavior of survival probability for a branching random walk with a barrier

Abstract

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+\varepsilon i^{\alpha} $ and all their descendants, where $ i $ presents the generation of the particles, $ \alpha > 1/3, \varepsilon\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 $ \varepsilon > 0. $ Moreover, we show that the decay of the probability is faster than $ e^{-\beta'\varepsilon^{\beta}} $ as $ \varepsilon\downarrow 0 $, where $ \beta', \beta $ are two positive constants depending on the branching random walk and $ \alpha $. 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.

Keywords

• branching random walk
• barrier
• survival probability