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), the barrier is set as ₙ+ n, where \ₙ\ 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 ₋ () ) of the particles system will be 0 (resp. , positive) when 0 (resp. , >0). In the present paper, we prove that L () will converge in Probability/ almost surely/ in Lᵖ to an explicit negative constant (depending on the environment) as 0 under some integrability conditions respectively. This result extends the scope of the result of Gantert et al. (2011) to the random environment case.