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, $\alpha >1/3,\epsilon \in \mathbb{R}$ 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 $\epsilon >0.$ Moreover, we show that the decay of the probability is faster than $e^{-{\beta }^{\prime }{\epsilon }^{\beta }}$ as $\epsilon \downarrow 0$, where ${\beta }^{\prime },\beta$ 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.