One-dimensional stepping stone models, sardine genetics and Brownian local time

• 成果类型: Article
• 署名作者: Durrett, Richard; Restrepo, Mateo
• 署名单位: Cornell University; Cornell University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/07-AAP451
• 发表日期: 2008
• 页码: 334-358
• 关键词: POPULATION

摘要

Consider a one-dimensional stepping stone model with colonies of size M and per-generation migration probability v, or a voter model on Z in which interactions occur over a distance of order K. Sample one individual at the origin and one at L. We show that if Mv/L and L/K-2 converge to positive finite limits, then the genealogy of the sample converges to a pair of Brownian motions that coalesce after the local time of their difference exceeds an independent exponentially distributed random variable. The computation of the distribution of the coalescence time leads to a one-dimensional parabolic differential equation with an interesting boundary condition at 0.