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December 2016 Genealogies in expanding populations
Rick Durrett, Wai-Tong (Louis) Fan
Ann. Appl. Probab. 26(6): 3456-3490 (December 2016). DOI: 10.1214/16-AAP1181

Abstract

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 [Theor. 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 rescaled 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.

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Rick Durrett. Wai-Tong (Louis) Fan. "Genealogies in expanding populations." Ann. Appl. Probab. 26 (6) 3456 - 3490, December 2016. https://doi.org/10.1214/16-AAP1181

Information

Received: 1 July 2015; Revised: 1 January 2016; Published: December 2016
First available in Project Euclid: 15 December 2016

zbMATH: 1358.92065
MathSciNet: MR3582808
Digital Object Identifier: 10.1214/16-AAP1181

Subjects:
Primary: 60H15 , 60K35
Secondary: 92C50

Keywords: Biased voter model , Duality , genealogies , Scaling limit , SPDE

Rights: Copyright © 2016 Institute of Mathematical Statistics

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Vol.26 • No. 6 • December 2016
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