Smooth density field 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: 1999
• 页码: 298-318
• 关键词: partial-differential equations; superprocesses; DIFFUSIONS; point

摘要

Given an (ordinary) super-Brownian motion (SBM) rho on R-d of dimension d = 2, 3, we consider a (catalytic) SBM X-rho on R-d with local branching rates rho(s)(dx). We show that X-t(rho) is absolutely continuous with a density function xi(t)(rho), say. Moreover, there exists a version of the map (t, z) --> xi(t)(rho)(z) which is l(infinity) and solves the heat equation off the catalyst rho; more precisely, off the (zero set of) closed support of the time-space measure ds rho(s)(dx). Using self-similarity we apply this result to give the following answer to an open problem on the long-term behavior of X-rho in dimension d = 2: if rho and X-rho Start with a Lebesgue measure, then does X-T(rho) converge (persistently) as T --> infinity toward a random multiple of Lebesgue measure?