Invariant measures of critical spatial branching processes in high dimensions

Abstract

We consider two critical spatial branching processes on $\mathbb{R}^d$: critical branching Brownian motion, and the critical Dawson-Watanabe process. A basic feature of these processes is that their ergodic behavior is highly dimension dependent. It is known that in low dimensions, $d \leq 2$, the only invariant measure is $\delta_0$ , the unit point mass on the empty state. In high dimensions, $d \geq 3$, there is a family ${\nu_{\theta}, \theta \epsilon [0, \infty)}$ of extremal invariant measures; the measures ${\nu_{\theta}$ are translation invariant and indexed by spatial intensity. We prove here, for $d \geq 3$, that all invariant measures are convex combinations of these measures.