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An asymptotic formula for the transition density of random genetic drift

Published online by Cambridge University Press:  14 July 2016

Peter L. Antonelli
Peter L. Antonelli*
Affiliation:
University of Alberta
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Abstract

Stochastic models in population genetics which lead to diffusion equations are considered. A geometric formula for the asymptotic expansions of the fundamental solutions of these equations is presented. Specifically, the random genetic drift process of one-locus theory and the Ohta–Kimura model of two-locus di-allelic systems with linkage are studied. Agreement with the work of Keller and Voronka for the two-allele one-locus case is obtained. For the general n-allele problem, the formulas obtained here are apparently new.

Keywords

WFK PROCESSRANDOM GENETIC DRIFTGEOMETRIC DIFFUSIONASYMPTOTIC TRANSITION DENSITYMUTATION FIELDRECOMBINATION FIELDCHRISTOFFEL VELOCITY FIELDRIEMANNIAN LOCALLY SYMMETRIC SPACERAY METHOD
Type
Short Communications
Information
Journal of Applied Probability , Volume 15 , Issue 1 , March 1978 , pp. 184 - 186
Copyright
Copyright © Applied Probability Trust 1978 

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References

[1] Antonelli, P. L. and Strobeck, C. (1977) The geometry of random drift, I. Stochastic distance and diffusion. Adv. Appl. Prob. 9, 238–249.CrossRefGoogle Scholar
[2] Antonelli, P. L., Morgan, K. and Lathrop, G. M. (1977) The geometry of random drift, III. Recombination and diffusion. Adv. Appl. Prob. 9, 260–267.Google Scholar
[3] Friedman, A. (1975) Stochastic Differential Equations and Applications, Vol. I. Academic Press, New York.Google Scholar
[4] Helgason, S. (1961) Differential Geometry and Symmetric Spaces. Academic Press, New York.Google Scholar
[5] Keller, J. B. and Voronka, R. (1975) Asymptotic analysis of stochastic models in population genetics. Math. Biosci. 25, 331–362.Google Scholar
[6] Molchanov, S. A. (1975) Diffusion processes and Riemannian geometry. Russian Math. Survey 30, 1–75.CrossRefGoogle Scholar