Finite time extinction of superprocesses with catalysts

• 成果类型: Article
• 署名作者: Dawson, DA; Fleischmann, K; Mueller, C
• 署名单位: Leibniz Association; Weierstrass Institute for Applied Analysis & Stochastics; University of Rochester
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 2000
• 页码: 603-642
• 关键词: super-brownian motion; local-times

摘要

Consider a catalytic super-Brownian motion X = X-Gamma with finite variance branching. Here catalytic means that branching of the reactant X is only possible in the presence of some catalyst. Our intrinsic example of a catalyst is a stable random measure Gamma on R of index 0 < < 1. Consequently, here the catalyst is located in a countable dense subset of R. Starting with a finite reactant mass Xo supported by a compact set, X is shown to die in finite time. We also deal with two other cases, with a power low catalyst and with a super-random walk on Z(d) with an i.i.d. catalyst. Our probabilistic argument uses the idea of good and bad historical paths of reactant particles during time periods [T-n, Tn+1). Good paths have a significant collision local time with the catalyst, and extinction can be shown by individual time change according to the collision local time and a comparison with Feller's branching diffusion. On the other hand, the remaining bad paths are shown to have a small expected mass at time Tn+1 which can be controlled by the hitting probability of point catalysts and the collision local time spent on them.