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November 2000 The biodiversity of catalytic super-Brownian motion
Klaus Fleischmann, Achim Klenke
Ann. Appl. Probab. 10(4): 1121-1136 (November 2000). DOI: 10.1214/aoap/1019487609

Abstract

In this paper we investigate the structure of the equilibriumstate of three-dimensional catalytic super-Brownian motion where the catalyst is itself a classical super-Brownian motion.We show that the reactant has an infinite local biodiversity or genetic abundance. This contrasts to the finite local biodiversity of the equilibriumof classical super-Brownian motion. Another question we address is that of extinction of the reactant in finite time or in the long-time limit in dimensions $d = 2,3$. Here we assume that the catalyst starts in the Lebesgue measure and the reactant starts in a finite measure.We show that there is extinction in the long-time limit if $d = 2 or 3$. There is, however, no finite time extinction if $d = 3$ (for $d = 2$, this problem is left open).This complements a result of Dawson and Fleischmann for $d = 1$ and again contrasts the behaviour of classical super-Brownian motion. As a key tool for both problems, we show that in $d = 3$ the reactant matter propagates everywhere in space immediately.

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Klaus Fleischmann. Achim Klenke. "The biodiversity of catalytic super-Brownian motion." Ann. Appl. Probab. 10 (4) 1121 - 1136, November 2000. https://doi.org/10.1214/aoap/1019487609

Information

Published: November 2000
First available in Project Euclid: 22 April 2002

zbMATH: 1073.60055
MathSciNet: MR1810867
Digital Object Identifier: 10.1214/aoap/1019487609

Subjects:
Primary: 60J80
Secondary: 60G57 , 60K35

Keywords: equilibrium states , extinction , genetic abundance , instantaneous propagation of matter , Superprocess

Rights: Copyright © 2000 Institute of Mathematical Statistics

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Vol.10 • No. 4 • November 2000
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