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

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 $\chi_n+an^{\alpha},$ where $a,\alpha$ are two constants and $\{\chi_n\}$ 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 $\alpha<1/3$ or $\alpha=1/3, a<a_c$ (resp., $\alpha>1/3, a>0$ or $\alpha=1/3, a>a_c$), where $a_c$ is a positive constant determined by the random environment. The rates of extinction when $\alpha<\frac{1}{3}, a\geq0$ and $\alpha=1/3, a\in(0,a_c)$ are also obtained. These extend the main results in A\"{\i}d\'{e}kon $\&$ Jaffuel (2011) and Jaffuel (2012),to the random environment case. The influence caused by the random environment have been specified.