ERGODICITY OF CRITICAL SPATIAL BRANCHING-PROCESSES IN LOW DIMENSIONS

• 成果类型: Article
• 署名作者: BRAMSON, M; COX, JT; GREVEN, A
• 署名单位: Syracuse University; University of Gottingen
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/aop/1176989006
• 发表日期: 1993
• 页码: 1946-1957
• 关键词: time; distributions; equation

摘要

We consider two critical spatial branching processes on R(d): critical branching Brownian motion, and the Dawson-Watanabe process. A basic feature of these processes is that their ergodic behavior is highly dimension-dependent. It is known that in low dimensions, d less than or equal to 2, the unique invariant measure with finite intensity is delta(0), the unit point mass on the empty state. In high dimensions, d greater than or equal to 3, there is a one-parameter family of nondegenerate invariant measures. We prove here that for d less than or equal to 2, delta(0) is the only invariant measure. In our proof we make use of sub- and super-solutions of the partial differential equation partial derivative u/partial derivative t = (1/2)Delta u - bu(2).