A note on the critical barrier for the survival of $\alpha-$stable branching random walk with absorption

TL;DR

This paper investigates the survival threshold of an $ ext{alpha}$-stable branching random walk with absorption, identifying a critical barrier level that determines whether the process persists or dies out, extending known results for the case when $ ext{alpha}=2$.

Contribution

It establishes the existence of a critical barrier for survival in $ ext{alpha}$-stable branching random walks, generalizing prior results from the Gaussian case to stable laws with $1< ext{alpha}<2$.

Findings

Existence of a critical barrier $a_ ext{alpha}$ for survival.

Survival occurs if the barrier exceeds $a_ ext{alpha}$.

Process dies out if the barrier is below $a_ ext{alpha}$.

Abstract

We consider a branching random walk with an absorbing barrier, where the step of the associated one-dimensional random walk is in the domain of attraction of an $\alpha$-stable law with $1<\alpha<2$. We shall prove that there is a barrier $an^{\frac{1}{1+\alpha}}$ and a critical value $a_\alpha$ such that if $a<a_\alpha$, then the process dies; if $a>a_\alpha$, then the process survives. The results generalize previous results in literature for the case $\alpha=2$.