GENEALOGIES IN EXPANDING POPULATIONS

• 成果类型: Article
• 署名作者: Durrett, Rick; Fan, Wai-Tong (Louis)
• 署名单位: Duke University; University of North Carolina; University of North Carolina Chapel Hill; University of North Carolina School of Medicine
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/16-AAP1181
• 发表日期: 2016
• 页码: 3456-3490
• 关键词: super-brownian motion; stochastic pdes; models; equation; contact; Duality; particles; selection; converge; front

摘要

The goal of this paper is to prove rigorous results for the behavior of genealogies in a one-dimensional long range biased voter model introduced by Hallatschek and Nelson [Thew: Pop. Biol. 73 (2008) 158-170]. The first step, which is easily accomplished using results of Mueller and Tribe [Probab. Theory Related Fields 102 (1995) 519-545], is to show that when space and time are resealed correctly, our biased voter model converges to a Wright-Fisher SPDE. A simple extension of a result of Durrett and Restrepo [Ann. Appl. Probab. 18 (2008) 334-358] then shows that the dual branching coalescing random walk converges to a branching Brownian motion in which particles coalesce after an exponentially distributed amount of intersection local time. Brunet et al. [Phys. Rev. E (3) 76 (2007) 041104,20] have conjectured that genealogies in models of this type are described by, the Bolthausen-Sznitman coalescent, see [Proc. Natl. Acad. Sci. USA 110 (2013) 437-442]. However, in the model we study there are no simultaneous coalescences. Our third and most significant result concerns tracer dynamics in which some of the initial particles in the biased voter model are labeled. We show that the joint distribution of the labeled and unlabeled particles converges to the solution of a system of stochastic partial differential equations. A new duality equation that generalizes the one Shiga [In Stochastic Processes in Physics and Engineering (Bielefeld, 1986) (1988) 345-355 Reidel] developed for the Wright-Fisher SPDE is the key to the proof of that result.