Open Access
October 1997 Multiple scale analysis of clusters in spatial branching models
Achim Klenke
Ann. Probab. 25(4): 1670-1711 (October 1997). DOI: 10.1214/aop/1023481107


In this paper we will investigate the long time behavior of critical branching Brownian motion and (finite variance) super-Brownian motion (the so-called Dawson-Watanabe process) on $\mathbb{R}$^d$. These processes are known to be persistent if $d \geq 3$; that is, there exist nontrivial equilibrium measures. If $d \leq 2$, they cluster; that is, the processes converge to the 0 configuration while the surviving mass piles up in so-called clusters.

We study the spatial profile of the clusters in the “critical” dimension $d = 2$ via multiple space scale analysis. We will also investigate the long-time behavior of these models restricted to finite boxes in $d \geq 2$. On the way, we develop coupling and comparison methods for spatial branching models.


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Achim Klenke. "Multiple scale analysis of clusters in spatial branching models." Ann. Probab. 25 (4) 1670 - 1711, October 1997.


Published: October 1997
First available in Project Euclid: 7 June 2002

zbMATH: 0909.60078
MathSciNet: MR1487432
Digital Object Identifier: 10.1214/aop/1023481107

Primary: 60J80
Secondary: 60G57 , 60K35

Keywords: Branching Brownian motion , cluster phenomena , Dawson-Watanabe (super) processes , finite systems

Rights: Copyright © 1997 Institute of Mathematical Statistics

Vol.25 • No. 4 • October 1997
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