The extinction rate of a branching random walk with a barrier in a time-inhomogeneous random environment

Consider a supercritical branching random walk in a time-inhomogeneous random environment. We impose a selection (called barrier) on survival in the following way. The position of the barrier may depend on the generation and the environment. %to indexed by time $n,$ i.e.,$\{{\text{Ł}}_n\}$with a random absorption barrier In each generation, only the individuals born below the barrier can survive and reproduce. When the barrier causes the extinction of the system, we give the extinction rate in the sense of $L^p(p\ge 1)$. Moreover, we show the $L^p$ convergence of the small deviation probability for a random walk with random environment in time.