An explicit transition density expansion for a multi-allelic Wright-Fisher diffusion with general diploid selection

TL;DR

This paper derives an explicit spectral expansion for the transition density of a multi-allelic Wright-Fisher diffusion with general diploid selection, enabling precise analysis of allele frequency dynamics.

Contribution

It extends a recent technique to obtain explicit transition densities for multi-allelic models with general diploid selection, including eigenvalues and eigenfunctions.

Findings

Explicit spectral representation of transition density.

Efficient computation of stationary distribution normalizing constant.

Accurate rate of convergence to equilibrium.

Abstract

Characterizing time-evolution of allele frequencies in a population is a fundamental problem in population genetics. In the Wright-Fisher diffusion, such dynamics is captured by the transition density function, which satisfies well-known partial differential equations. For a multi-allelic model with general diploid selection, various theoretical results exist on representations of the transition density, but finding an explicit formula has remained a difficult problem. In this paper, a technique recently developed for a diallelic model is extended to find an explicit transition density for an arbitrary number of alleles, under a general diploid selection model with recurrent parent-independent mutation. Specifically, the method finds the eigenvalues and eigenfunctions of the generator associated with the multi-allelic diffusion, thus yielding an accurate spectral representation of the transition density. Furthermore, this approach allows for efficient, accurate computation of various other quantities of interest, including the normalizing constant of the stationary distribution and the rate of convergence to this distribution.