The Annals of Probability

Multiple scale analysis of clusters in spatial branching models

Achim Klenke

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Abstract

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.

Article information

Source
Ann. Probab., Volume 25, Number 4 (1997), 1670-1711.

Dates
First available in Project Euclid: 7 June 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1023481107

Digital Object Identifier
doi:10.1214/aop/1023481107

Mathematical Reviews number (MathSciNet)
MR1487432

Zentralblatt MATH identifier
0909.60078

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60K35 60G57: Random measures

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

Citation

Klenke, Achim. Multiple scale analysis of clusters in spatial branching models. Ann. Probab. 25 (1997), no. 4, 1670--1711. doi:10.1214/aop/1023481107. https://projecteuclid.org/euclid.aop/1023481107


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