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

TL;DR

This paper investigates the survival and extinction of a supercritical branching random walk in a time-inhomogeneous random environment with an absorption barrier, identifying critical parameters that determine the process's fate.

Contribution

It extends previous results on branching random walks to the case of random environments, characterizing the impact of environment randomness on survival thresholds.

Findings

Survival occurs if lpha>1/3 or lpha=1/3 with a>a_c.

Extinction occurs if lpha<1/3 or lpha=1/3 with a<a_c.

Explicit extinction rates are derived for certain parameter regimes.

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 $\chi_n+an^{\alpha},$ 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.