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

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+epsilon i alpha and all their descendants, where i presents the generation of the particles, alpha > 1/3, epsilon is an element of 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 epsilon > 0. Moreover, we show that the decay of the probability is faster than e-beta'epsilon beta as epsilon down arrow 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.