Recurrence and capacity of stable branching random walks

Source abstract

We study the linear growth rate of the range of size-conditioned Branching Random Walks (BRW) when the offspring distribution $μ$ is critical and attracted to an $α$-stable law. This is done via the infinite invariant BRW introduced by Le Gall & Lin and a new criterion which relates this growth rate of the range to a notion of dimension of the underlying tree in a general way. Then, in the transient case (that is, when the range does grow linearly), we extend the notion of branching capacity to this $α$-stable case. We show that it is still related to the asymptotic probability that a BRW (or its infinite version) reaches a distant set in $\mathbb Z^d$, and we estimate the $α$-stable branching capacity of balls.