Asymptotic Green's function for the stochastic reproduction of competing variants via Fisher's angular transformation

Abstract

The Wright-Fisher Fokker-Planck equation describes the stochastic dynamics of self-reproducing, competing variants at fixed population size. We use Fisher's angular transformation, which defines a natural length for this stochastic process, to remove the co-ordinate dependence of it's diffusive dynamics, resulting in simple Brownian motion in an unstable potential, driving variants to extinction or fixation. This insight allows calculation of very accurate asymptotic formula for the Green's function under neutrality and selection, using a novel heuristic Gaussian approximation.