On the barrier problem of branching random walk in time-inhomogeneous random environment

Abstract

We consider a supercritical branching random walk in time-inhomogeneous random environment with a random absorption barrier, i.e.,in each generation, only the individuals born below the barrier can survive and reproduce. Assume that the random environment is i.i.d..The barrier is set as $\ddot{I}{‡}_n+a{n}^{\hat{I}}\pm ,$ where $a,\hat{I}\pm$ are two constants and $\{\ddot{I}{‡}_n\}$ is a certain i.i.d. random walk determined by the random environment.We show that for almost surely given environment (i.e., a sequence of point processes which is a realization of the random environment), the time-inhomogeneous branching random walk under the given environment will become extinct (resp., survive with positive probability) if $\hat{I}\pm <1/3$ or $\hat{I}\pm =1/3,a<a_c$ (resp., $\hat{I}\pm >1/3,a>0$ or $\hat{I}\pm =1/3,a>a_c$), where $a_c$ is a positive constant determined by the random environment. The rates of extinction when $\hat{I}\pm <\frac{1}{3},a\ge 0$ and $\hat{I}\pm =1/3,a\in (0,a_c)$ are also obtained. These extend the main results in A\"ıdékon $\&$ Jaffuel (2011) and Jaffuel (2012),to the random environment case. The influence caused by the random environment have been specified.