Birth and death process with one-side bounded jumps in random environment

Abstract

Let $ω=(ω_i)_{i\in\mathbb Z}=(μ^{L}_i,...,μ^{1}_i,λ_i)_{i\in \mathbb Z}$, which serves as the environment, be a sequence of i.i.d. random nonnegative vectors, with $L\ge1$ a positive integer. We study birth and death process $N_t$ which, given the environment $ω,$ waits at a state $n$ an exponentially distributed time with parameter $λ_n+\sum_{l=1}^Lμ^{l}_n$ and then jumps to $n-i$ with probability ${μ^i_n}/(λ_n+\sum_{l=1}^Lμ^{l}_n),$ $i=1,...,L$ or to $n+1$ with probability ${λ_n}/(λ_n+\sum_{l=1}^Lμ^{l}_n).$ A sufficient condition for the existence, a criterion for recurrence, and a law of large numbers of the process $N_t$ are presented. We show that the first passage time $T_1\overset{\mathscr D}{=}ξ_{0,1}+\sum_{i\le -1}\sum_{k=1}^{U_{i,1}}ξ_{i,k}+\sum_{i\le -1}\sum_{k= 1}^{U_{i,1}+...+U_{i,L}}\tildeξ_{i+1,k},$ where $(U_{i,1},...,U_{i,L})_{i\le0}$ is an $L$-type branching process in random environment and, given $ω,$ $ξ_{i,k},\ \tildeξ_{i,k},\ i\le 0,\ k\ge 1$ are mutually independent random variables such that $P_ω(ξ_{i,k}\ge t)=e^{-(λ_i+\sum_{l=1}^Lμ^{l}_i)t},\ t\ge 0.$ This fact enables us to give an explicit velocity of the law of large numbers.