Invariant measures of critical spatial branching processes in high dimensions

• 成果类型: Article
• 署名作者: Bramson, M; Cox, JT; Greven, A
• 署名单位: University of Wisconsin System; University of Wisconsin Madison; Syracuse University; University of Erlangen Nuremberg
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 1997
• 页码: 56-70
• 关键词: Diffusions; SYSTEM; time

摘要

We consider two critical spatial branching processes on R-d: critical branching Brownian motion, and the critical 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 only invariant measure is delta(0), the unit point mass on the empty state. In high dimensions, d greater than or equal to 3, there is a family {nu(0), theta is an element of [0, infinity)} of extremal invariant measures; the measures nu(0) are translation invariant and indexed by spatial intensity. We prove here, for d greater than or equal to 3, that all invariant measures are convex combinations of these measures.