POISSONIAN STATISTICS IN THE EXTREMAL PROCESS OF BRANCHING BROWNIAN MOTION

• 成果类型: Article
• 署名作者: Arguin, Louis-Pierre; Bovier, Anton; Kistler, Nicola
• 署名单位: Universite de Montreal; University of Bonn
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/11-AAP809
• 发表日期: 2012
• 页码: 1693-1711
• 关键词: equation; wave

摘要

As a first step toward a characterization of the limiting extremal process of branching Brownian motion, we proved in a recent work [Comm. Pure Appl. Math. 64 (2011) 1647-1676] that, in the limit of large time t, extremal particles descend with overwhelming probability from ancestors having split either within a distance of order 1 from time 0, or within a distance of order 1 from time t. The result suggests that the extremal process of branching Brownian motion is a randomly shifted cluster point process. Here we put part of this picture on rigorous ground: we prove that the point process obtained by retaining only those extremal particles which are also maximal inside the clusters converges in the limit of large t to a random shift of a Poisson point process with exponential density. The last section discusses the Tidal Wave Conjecture by Lalley and Sellke [Ann. Probab. 15 (1987) 1052-1061] on the full limiting extremal process and its relation to the work of Chauvin and Rouault [Math. Nachr 149 (1990) 41-59] on branching Brownian motion with atypical displacement.