The Annals of Probability

Ergodicity of Critical Spatial Branching Processes in Low Dimensions

Maury Bramson, J. T. Cox, and Andreas Greven

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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 $\partial u/\partial t = (1/2) \Delta u - bu^2$.

Article information

Ann. Probab., Volume 21, Number 4 (1993), 1946-1957.

First available in Project Euclid: 19 April 2007

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Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Critical branching Brownian motion Dawson-Watanabe process invariant measures


Bramson, Maury; Cox, J. T.; Greven, Andreas. Ergodicity of Critical Spatial Branching Processes in Low Dimensions. Ann. Probab. 21 (1993), no. 4, 1946--1957. doi:10.1214/aop/1176989006.

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