Invariant measures of critical branching random walks in high dimension

In this work, we characterize cluster-invariant point processes for critical branching spatial processes on Rd 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 α≤2 and the offspring law μ of the branching process has an heavy tail such that μ(k)∼k−2−β, then we need the dimension d to be strictly larger than the critical dimension α∕β. 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.