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 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 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 [BCG97] whose proof used PDE techniques, our proof uses probabilistic tools only.