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October, 1993 Ergodicity of Critical Spatial Branching Processes in Low Dimensions
Maury Bramson, J. T. Cox, Andreas Greven
Ann. Probab. 21(4): 1946-1957 (October, 1993). DOI: 10.1214/aop/1176989006


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


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Maury Bramson. J. T. Cox. Andreas Greven. "Ergodicity of Critical Spatial Branching Processes in Low Dimensions." Ann. Probab. 21 (4) 1946 - 1957, October, 1993.


Published: October, 1993
First available in Project Euclid: 19 April 2007

zbMATH: 0788.60119
MathSciNet: MR1245296
Digital Object Identifier: 10.1214/aop/1176989006

Primary: 60K35
Secondary: 60J80

Keywords: Critical branching Brownian motion , Dawson-Watanabe process , Invariant measures

Rights: Copyright © 1993 Institute of Mathematical Statistics

Vol.21 • No. 4 • October, 1993
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