The Annals of Applied Probability
- Ann. Appl. Probab.
- Volume 18, Number 1 (2008), 334-358.
One-dimensional stepping stone models, sardine genetics and Brownian local time
Consider a one-dimensional stepping stone model with colonies of size M and per-generation migration probability ν, 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 Mν/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 one-dimensional parabolic differential equation with an interesting boundary condition at 0.
Ann. Appl. Probab., Volume 18, Number 1 (2008), 334-358.
First available in Project Euclid: 9 January 2008
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Durrett, Richard; Restrepo, Mateo. One-dimensional stepping stone models, sardine genetics and Brownian local time. Ann. Appl. Probab. 18 (2008), no. 1, 334--358. doi:10.1214/07-AAP451. https://projecteuclid.org/euclid.aoap/1199890025