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January 1997 Invariant measures of critical spatial branching processes in high dimensions
Maury Bramson, J. T. Cox, Andreas Greven
Ann. Probab. 25(1): 56-70 (January 1997). DOI: 10.1214/aop/1024404278


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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Maury Bramson. J. T. Cox. Andreas Greven. "Invariant measures of critical spatial branching processes in high dimensions." Ann. Probab. 25 (1) 56 - 70, January 1997.


Published: January 1997
First available in Project Euclid: 18 June 2002

zbMATH: 0882.60091
MathSciNet: MR1428499
Digital Object Identifier: 10.1214/aop/1024404278

Primary: 60K35
Secondary: 60J80

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

Rights: Copyright © 1997 Institute of Mathematical Statistics

Vol.25 • No. 1 • January 1997
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