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Ergodicity of critical spatial branching processes in low dimensions. (English) Zbl 0788.60119

Summary: We consider two critical spatial branching processes on \(\mathbb{R}^ d\): critical branching Brownian motion, and the 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 unique invariant measure with finite intensity is \(\delta_ 0\), the unit point mass on the empty state. In high dimensions, \(d\geq 3\), there is a one-parameter family of nondegenerate invariant measures. We prove here that for \(d\leq 2\), \(\delta_ 0\) is the only invariant measure. In our proof we make use of sub- and super-solutions of the partial differential equation \(\partial u/\partial t = (1/2)\Delta u - bu^ 2\).

MSC:

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