The Annals of Applied Probability

A transition function expansion for a diffusion model with selection

A. D. Barbour, S. N. Ethier, and R. C. Griffiths

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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.

Article information

Source
Ann. Appl. Probab., Volume 10, Number 1 (2000), 123-162.

Dates
First available in Project Euclid: 25 April 2002

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1019737667

Digital Object Identifier
doi:10.1214/aoap/1019737667

Mathematical Reviews number (MathSciNet)
MR1765206

Zentralblatt MATH identifier
1171.60368

Subjects
Primary: 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07] 60J60: Diffusion processes [See also 58J65] 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 92D10: Genetics {For genetic algebras, see 17D92}

Keywords
Finite-dimensional diffusion process population genetics duality reversibility multitype birth-and-death process coupling Ray-Knight compactification

Citation

Barbour, A. D.; Ethier, S. N.; Griffiths, R. C. A transition function expansion for a diffusion model with selection. Ann. Appl. Probab. 10 (2000), no. 1, 123--162. doi:10.1214/aoap/1019737667. https://projecteuclid.org/euclid.aoap/1019737667


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