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

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.

arXiv comment28 pages, 12 figures

References in corpus (4)

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• Exact Solution of the Multi-Allelic Diffusion Model
• Multivariate Jacobi and Laguerre polynomials, infinite-dimensional extensions, and their probabilistic connections with multivariate Hahn and Meixner polynomials
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Cited by in corpus (10)

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• Analysis and rejection sampling of Wright-Fisher diffusion bridges
• Polygenic dynamics underlying the response of quantitative traits to directional selection
• Diffusion processes satisfying a conservation law constraint
• Addendum to: Bounds for survival probabilities in supercritical Galton-Watson processes and applications to population genetics
• Bounds for survival probabilities in supercritical Galton-Watson processes and applications to population genetics
• A stochastic diffusion process for Lochner's generalized Dirichlet distribution
• On recurrent properties of Fisher-Wright's diffusion on $(0,1)$ with mutation