On the hot spots of a catalytic super-Brownian motion

• 成果类型: Article
• 署名作者: Delmas, JF; Fleischmann, K
• 署名单位: Institut Polytechnique de Paris; Ecole Nationale des Ponts et Chaussees; Leibniz Association; Weierstrass Institute for Applied Analysis & Stochastics
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s004400100156
• 发表日期: 2001
• 页码: 389-421
• 关键词: point catalyst; DIFFUSIONS; time

摘要

Consider the catalytic super-Brownian motion X-rho (reactant) in R-d, d less than or equal to 3, which branching rates vary randomly in time and space and in fact are given by an ordinary super-Brownian motion rho (catalyst). Our main object of study is the collision local time L = L-[rho ,L-Xo](d(s, x)) of catalyst and reactant. It determines the covariance measure in the martingale problem for X-rho and reflects the occurrence of hot spots of reactant which can be seen in simulations of X-rho. In dimension 2, the collision local time is absolutely continuous in time, L(d(s, x)) = ds K-s (dx). At fixed time s, the collision measures K-rho (dx) of rho (s) and XO have carrying Hausdorff dimension 2. Spatial marginal densities of L exist, and, via self-similarity, enter in the long-term random ergodic limit of L (diffusiveness of the 2-dimensional model). We also compare some of our results with the case of super-Brownian motions with deterministic time-independent catalysts.