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

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 $a{n}^{\frac{1}{1+\alpha }}$ 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 $\alpha =2$.