The biodiversity of catalytic super-Brownian motion

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=2or3. 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.