Asymptotic behavior for the quenched survival probability of a supercritical branching random walk in random environment with a barrier

Abstract

We introduce a random barrier to a supercritical branching random walk in an i.i.d. random environment $\{{\mathcal{L}}_n\}$ indexed by time $n,$ i.e., in each generation, only the individuals born below the barrier can survive and reproduce. At generation $n$ ($n\in \mathbb{N}$), the barrier is set as ${\chi }_n+\epsilon n,$ where $\{{\chi }_n\}$ is a random walk determined by the random environment. Lv \& Hong (2024) showed that for almost every $\mathcal{L}:=\{{\mathcal{L}}_n\},$ the quenched survival probability (denoted by ${\varrho }_{\mathcal{L}}(\epsilon )$) of the particles system will be 0 (resp., positive) when $\epsilon \le 0$ (resp., $\epsilon >0$). In the present paper, we prove that $\sqrt{\epsilon }\log {\varrho }_{\mathcal{L}}(\epsilon )$ will converge in Probability/ almost surely/ in $L^p$ to an explicit negative constant (depending on the environment) as $\epsilon \downarrow 0$ under some integrability conditions respectively. This result extends the scope of the result of Gantert et al. (2011) to the random environment case.