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The biodiversity of catalytic super-Brownian motion. (English) Zbl 1073.60055

Summary: We investigate the structure of the equilibrium state 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 equilibrium of 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.

MSC:

60G57 Random measures
60J65 Brownian motion
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60K35 Interacting random processes; statistical mechanics type models; percolation theory
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