Abstract
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 $\partial u/\partial t = (1/2) \Delta u - bu^2$.
Citation
Maury Bramson. J. T. Cox. Andreas Greven. "Ergodicity of Critical Spatial Branching Processes in Low Dimensions." Ann. Probab. 21 (4) 1946 - 1957, October, 1993. https://doi.org/10.1214/aop/1176989006
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