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Invariant measures of critical spatial branching processes in high dimensions. (English) Zbl 0882.60091

Summary: 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 \(\{\nu_\theta,\;\theta\in [0,\infty)\}\) 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.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
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