HAUSDORFF MEASURE OF ARCS AND BROWNIAN MOTION ON BROWNIAN SPATIAL TREES

• 成果类型: Article
• 署名作者: Croydon, David A.
• 署名单位: University of Warwick
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/08-AOP425
• 发表日期: 2009
• 页码: 946-978
• 关键词: continuum random tree; CONVERGENCE; GROWTH

摘要

A Brownian spatial tree is defined to be a pair (T, phi), where T is the rooted real tree naturally associated with a Brownian excursion and phi is a random continuous function from T into R(d) such that, conditional on T, phi maps each arc of T to the image of a Brownian motion path in R(d) run for a time equal to the arc length. It is shown that, in high dimensions, the Hausdorff measure of arcs can be used to define an intrinsic metric d(s) on the set S := phi(T). Applications of this result include the recovery of the spatial tree (T, phi) from the set s alone, which implies in turn that a Dawson-Watanabe super-process can be recovered from its range. Furthermore, d(s) can be used to construct a Brownian motion on s, which is proved to be the scaling limit of simple random walks on related discrete structures. In particular, a limiting result for the simple random walk on the branching random walk is obtained.