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 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.