Math @ Duke

Publications [#243553] of Richard T. Durrett
Papers Published
 Durrett, R; Restrepo, M, Onedimensional stepping stone models, sardine genetics and Brownian local time,
The annals of applied probability : an official journal of the Institute of Mathematical Statistics, vol. 18 no. 1
(2008),
pp. 334358, ISSN 10505164 [MR2380901 (2008j:60229)], [doi]
(last updated on 2017/12/16)
Abstract: Consider a onedimensional stepping stone model with colonies of size M and pergeneration 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/K2 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 onedimensional parabolic differential equation with an interesting boundary condition at 0. © Institute of Mathematical Statistics, 2008.


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