Invariant measures of critical branching random walks in high dimension

Abstract

In this work, we characterize cluster-invariant point processes for critical branching spatial processes on ${\mathbb{R}}^d$ for all large enough d when the motion law is α-stable or has a finite discrete range. More precisely, when the motion is α-stable with $\mathit{\alpha }\le 2$ and the offspring law μ of the branching process has an heavy tail such that $\mathrm{\mu }(k)\sim k^{-2-\mathit{\beta }}$, then we need the dimension d to be strictly larger than the critical dimension $\mathit{\alpha }∕\mathit{\beta }$. In particular, when the motion is Brownian and the offspring law μ has a second moment, this critical dimension is 2. Contrary to the previous work of Bramson, Cox and Greven in [4] whose proof used PDE techniques, our proof uses probabilistic tools only.