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

We introduce a random barrier to a supercritical branching random walk in an i.i.d. random environment {Ln} indexed by time n, i.e., in each generation, only the individuals born below the barrier can survive and reproduce. At generation n (n∈N), the barrier is given by χn+εn, where {χn} is a random walk determined by the random environment. Lv & Hong (2024) showed that for almost every L:={Ln}, the quenched survival probability (denoted by ϱL(ε)) of the particle system will be 0 (resp., positive) when ε≤0 (resp., ε>0). In the present paper, we prove that εlogϱ L(ε) will converge in Probability/ almost surely/ in Lp to an explicit negative constant (depending on the environment) as ε↓0 under some integrability conditions respectively. This result extends the scope of the result of Gantert et al. (2011) to the random environment case.