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 & alpha;-stable or has a finite discrete range. More precisely, when the motion is & alpha;-stable with & alpha; & LE; 2 and the offspring law & mu; of the branching process has an heavy tail such that & mu;(k) & SIM; k-2-& beta;, then we need the dimension d to be strictly larger than the critical dimension & alpha;/& beta;. In particular, when the motion is Brownian and the offspring law & mu; 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.

Automated summary

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. Our proof uses probabilistic tools only, contrary to the previous work that used PDE techniques.