The Annals of Probability

Invariant measures of critical spatial branching processes in high 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 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 ${\nu_{\theta}, \theta \epsilon [0, \infty)}$ 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.

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Ann. Probab., Volume 25, Number 1 (1997), 56-70.

First available in Project Euclid: 18 June 2002

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Zentralblatt MATH identifier

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 critical Dawson-Watanabe process invariant measures


Bramson, Maury; Cox, J. T.; Greven, Andreas. Invariant measures of critical spatial branching processes in high dimensions. Ann. Probab. 25 (1997), no. 1, 56--70. doi:10.1214/aop/1024404278.

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