Branching exit Markov systems and superprocesses

• 成果类型: Article
• 署名作者: Dynkin, EB
• 署名单位: Cornell University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/aop/1015345774
• 发表日期: 2001
• 页码: 1833-1858
• 关键词: differential-equations

摘要

Superprocesses (under the name continuous state branching processes) appeared, first, in a pioneering work of S. Watanabe [J. Math.. Kyoto Univ. 8 (1968) 141-167]. Deep results on paths of the super-Brownian motion were obtained by Dawson, Perkins, Le Gall and others. In earlier papers, a superprocess was interpreted as a Markov process X-t in the space of measures. This is not sufficient for a probabilistic approach to boundary value problems. A reacher model based on the concept of exit measures was introduced by E. B. Dynkin [Probab. Theory Related Fields 89 (1991) 89-115]. A model of a superprocess as a system of exit measures from time-space open sets was systematically developed in 1993 [E. B. Dynkin, Ann. Probab. 21 (1993) 1185-1262]. In particular; branching and Markov properties of such a system were established and used to investigate partial differential equations. In the present paper; we show that the entire theory of superprocesses can be deduced from these properties.