The biodiversity of catalytic super-Brownian motion

• 成果类型: Article
• 署名作者: Fleischmann, K; Klenke, A
• 署名单位: Leibniz Association; Weierstrass Institute for Applied Analysis & Stochastics; University of Erlangen Nuremberg
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• 发表日期: 2000
• 页码: 1121-1136
• 关键词: superprocesses

摘要

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 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.