On the maximal displacement of subcritical branching random walk in random environment

TL;DR

This paper analyzes the maximum displacement of subcritical branching random walks in random environments, focusing on probabilistic properties and the effects of environmental randomness on particle spread.

Contribution

It provides new insights into the maximal displacement behavior of subcritical branching random walks under random environmental conditions.

Findings

Derived bounds for maximal displacement in subcritical regimes

Established probabilistic limit theorems for displacement

Demonstrated the impact of environmental randomness on particle spread

Abstract

In this paper, we consider the subcritical branching random walk in a random environment. We assume the branching and the step jump are independent; and the branching is in random envirenment, i.e., the particles in generation $n$ produce children according the probability measure $F_n\in \mathcal{P}\left(\N_0\right)$, and the $F_n$, $n=1,2,\cdots, $ are i.i.d under the $P_E$. ``subcritical" means that $ a:=\E[X_1]\in (-\infty,0)$, where $X_1:=\log \overline{F}_1$ and $\overline{F}_1$ is the mean of $F_1$.