Invariant measures of critical branching random walks in high dimension

TL;DR

This paper characterizes invariant measures for critical branching random walks in high dimensions, focusing on stable and finite-range motions, and provides a probabilistic proof contrasting previous PDE-based approaches.

Contribution

It offers a probabilistic characterization of cluster-invariant point processes for critical branching spatial processes in high dimensions, extending prior work with new techniques.

Findings

Invariant measures depend on the dimension being larger than a critical value.

For Brownian motion with finite variance offspring, the critical dimension is 2.

Probabilistic methods are used instead of PDE techniques.

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 $\alpha$-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 [BCG97] whose proof used PDE techniques, our proof uses probabilistic tools only.