THE UNIFORM RANDOM TREE IN A BROWNIAN EXCURSION

• 成果类型: Article
• 署名作者: LEGALL, JF
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/BF01292678
• 发表日期: 1993
• 页码: 369-383
• 关键词: branching-processes; motion

摘要

To any Brownian excursion e with duration sigma(e) and any t1, ..., t(p) is-an-element-of[0, sigma(e)], we associate a branching tree with p branches denoted by T(p)(e, t1, ..., t(p)), which is closely related to the structure of the minima of e. Our main theorem states that, if e is chosen according to the Ito measure and (t1, ..., t(p)) according to Lebesgue measure on [0, sigma(e)]p, the tree T(p)(e, t1, ..., t(p)) is distributed according to the uniform measure on the set of trees with p branches. The proof of this result yields additional information about the '' subexcursions'' of e corresponding to the different branches of the tree, thus generalizing a well-known representation theorem of Bismut. If we replace the Ito measure by the law of the normalized excursion, a simple conditioning argument leads to another remarkable result originally proved by Aldous with a very different method.