# A transition function expansion for a diffusion model with selection - Zurich Open Repository and Archive

Barbour, A D; Ethier, S; Griffiths, R (2000). A transition function expansion for a diffusion model with selection. Annals of Applied Probability, 10(1):123-162.

## Abstract

Using duality, an expansion is found for the transition function of the reversible $K$-allele diffusion model in population genetics. In the neutral case, the expansion is explicit but already known. When selection is present, it depends on the distribution at time $t$ of a specified $K$-type birth-and-death process starting at "infinity". The latter process is constructed by means of a coupling argument and characterized as the Ray process corresponding to the Ray–Knight compactification of the $K$-dimensional nonnegative-integer lattice.

## Abstract

Using duality, an expansion is found for the transition function of the reversible $K$-allele diffusion model in population genetics. In the neutral case, the expansion is explicit but already known. When selection is present, it depends on the distribution at time $t$ of a specified $K$-type birth-and-death process starting at "infinity". The latter process is constructed by means of a coupling argument and characterized as the Ray process corresponding to the Ray–Knight compactification of the $K$-dimensional nonnegative-integer lattice.

## Citations

14 citations in Web of Science®
16 citations in Scopus®

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Item Type: Journal Article, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics Finite-dimensional diffusion process; population genetics; duality; reversibility; multitype birth-and-death process; coupling; Ray-Knight compactification English 2000 07 Apr 2010 12:45 05 Apr 2016 13:25 Institute of Mathematical Statistics 1050-5164 https://doi.org/10.1214/aoap/1019737667