Biodiversity of Catalytic Super-Brownian Motion

In this paper 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 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 (1997a) 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 = 3the reactant matter propagates everywhere in space immediately