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

TL;DR

This paper derives an asymptotic Green's function for the Wright-Fisher process using Fisher's angular transformation, simplifying the stochastic dynamics of competing variants and enabling precise calculations of fixation probabilities.

Contribution

It introduces a novel heuristic Gaussian approximation to compute the Green's function for the Wright-Fisher process under neutrality and selection.

Findings

Accurate asymptotic Green's function derived for the Wright-Fisher process.

Simplification of stochastic dynamics via Fisher's angular transformation.

Enhanced ability to calculate fixation probabilities and times.

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.