Invariant measures of critical branching random walks in high dimension

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