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

We introduce a random absorption barrier to a supercritical branching random walk with an i.i.d. random environment {L-n} indexed by time n, i.e., in each generation, only the individuals born below the barrier can survive and reproduce. The barrier is set as chi(n)+ an(alpha), where a, alpha are two constants and {chi(n)} is a random walk determined by the random environment. We show that for almost every L := {L-n}, the time-inhomogeneous branching random walk with barrier 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 ac is a positive constant determined by the random environment. The rates of extinction when a < 1 3, a >= 0 and alpha = 1/3, a is an element of (0, a(c)) are also obtained. These extend the main results in Aidekon and Jaffuel (2011) and Jaffuel (2012) to the random environment case. The influence of the random environment has been specified.