The Annals of Probability

Clustering and invariant measures for spatial branching models with infinite variance

Achim Klenke

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Abstract

We consider two spatial branching models on $\mathbb{R}^d$: branching Brownian motion with a branching law in the domain of normal attraction of a $1 + \beta$ stable law, $0 < \beta \leq 1$, and the corresponding high density limit measure valued diffusion. The longtime behavior of both models depends highly on $\beta$ and $d$. We show that for $d \leq \frac{2}{\beta}$ the only invariant measure is $\delta_0$, the unit mass on the empty configuration. Furthermore, we give a precise condition for convergence toward $\delta_0$. For $d > \frac{2}{\beta}$ it is known that there exists a family $(\nu_{\theta}, \theta \epsilon [0, \infty))$ of nontrivial invariant measures. We show that every invariant measure is a convex combination of the $\nu_{\theta}$. Both results have been known before only under an additional finite mean assumption. For the critical dimension $d = \frac{2}{\beta}$ we show that both models display the phenomenon of diffusive clustering. This means that clusters grow spatially on a random scale. We give a precise description of the clusters via multiple scale analysis. Our methods rely mainly on studying sub- and supersolutions of the reaction diffusion equation $\frac{\partial u}{\partial t} - 1/2 \Delta u + u^{1 + \beta} = 0$.

Article information

Source
Ann. Probab., Volume 26, Number 3 (1998), 1057-1087.

Dates
First available in Project Euclid: 31 May 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1022855745

Digital Object Identifier
doi:10.1214/aop/1022855745

Mathematical Reviews number (MathSciNet)
MR1634415

Zentralblatt MATH identifier
0938.60086

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60G57: Random measures

Keywords
Branching Brownian motion superprocess invariant measures diffusive clustering stable laws reaction-diffusion equation

Citation

Klenke, Achim. Clustering and invariant measures for spatial branching models with infinite variance. Ann. Probab. 26 (1998), no. 3, 1057--1087. doi:10.1214/aop/1022855745. https://projecteuclid.org/euclid.aop/1022855745


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