THE GENEALOGY OF BRANCHING BROWNIAN MOTION WITH ABSORPTION

• 成果类型: Article
• 署名作者: Berestycki, Julien; Berestycki, Nathanael; Schweinsberg, Jason
• 署名单位: Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Sorbonne Universite; Universite Paris Cite; University of Cambridge; University of California System; University of California San Diego
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/11-AOP728
• 发表日期: 2013
• 页码: 527-618
• 关键词: survival probability; limit-theorem; random-walk; particle-systems; traveling-waves; coalescent; equation; displacement; asymptotics; selection

摘要

We consider a system of particles which perform branching Brownian motion with negative drift and are killed upon reaching zero, in the near-critical regime where the total population stays roughly constant with approximately N particles. We show that the characteristic time scale for the evolution of this population is of order (log N)(3), in the sense that when time is measured in these units, the scaled number of particles converges to a variant of Neveu's continuous-state branching process. Furthermore, the genealogy of the particles is then governed by a coalescent process known as the Bolthausen-Sznitman coalescent. This validates the nonrigorous predictions by Brunet, Derrida, Muller and Munier for a closely related model.