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An application of time reversal of Markov processes to a problem of population genetics

Published online by Cambridge University Press:  01 July 2016

Masao Nagasawa  and
Takeo Maruyama
Masao Nagasawa
Affiliation:
Tokyo Institute of Technology
Takeo Maruyama*
Affiliation:
National Institute of Genetics, Mishima
*
∗∗Postal address: National Institute of Genetics, Mishima, Shizuoka, Japan.
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Abstract

A formula is proved for the expected value of sum of a function of gene frequency along a sample path in the past, given the present frequency. The proof and the explanation of the formula is based on the general theory of time reversal of Markov processes. Moreover, a relation between time reversal and conditional processes is discussed, and it is shown that the fictitious drift term appears when one looks back at the history of mutants given the present frequency.

Keywords

POPULATION GENETICSDIFFUSION PROCESSESTIME REVERSAL OF MARKOV PROCESSESCONDITIONAL SOJOURN TIMES OF GENE FREQUENCYCONDITIONAL DIFFUSIONSFICTITIOUS DRIFT
Type
Research Article
Information
Advances in Applied Probability , Volume 11 , Issue 3 , September 1979 , pp. 457 - 478
Copyright
Copyright © Applied Probability Trust 1979 

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Footnotes

∗

Present address: Seminar für Angewandte Mathematik der Universität Zürich, Freiestrasse 36, CH-8032 Zürich, Switzerland.

References

Feller, W. (1954) Diffusion processes in one dimension. Trans. Amer. Math. Soc. 77, 1–31.CrossRefGoogle Scholar
Freire-Maia, A., Li, W.-H. and Maruyama, T. (1975) Genetics of Acheiropodia (the handless and footless families of Brazil). VII. Population dynamics. Amer. J. Hum. Genet. 27, 665–675.Google ScholarPubMed
Ito, K. (1957) Kakuritzu Katei. Iwanami Shoten, Tokyo. English translation: Ito, Y. (1961) Last two chapters of Kiyosi Ito's monograph Stochastic Processes. Department of Mathematics, Yale University.Google Scholar
Ito, K. and McKean, H. P. (1965) Diffusion Processes and their Sample Paths. Springer-Verlag, Berlin.Google Scholar
Kolmogoroff, A. (1937) Zur Umkehrbarkeit der statistischen Naturgesetze. Math. Ann. 113, 766–772.CrossRefGoogle Scholar
Levikson, B. (1977) The age distribution of Markov processes. J. Appl. Prob. 14, 492–506.Google Scholar
Maruyama, T. (1977) Stochastic Problems in Population Genetics. Lecture Notes in Biomathematics 17, Springer-Verlag, Berlin.Google Scholar
Maruyama, , T. and Kimura, M. (1975) Moments for sum of an arbitrary function of gene frequency along a stochastic path of gene frequency change. Proc. Nat. Acad. Sci. U.S.A. 72, 1602–1604.Google Scholar
Meyer, , P. A., Smythe, R. T. and Walsh, J. B. (1972) Birth and death of Markov processes. Proc. 6th Berkeley Symp. Math. Statist. Prob. 3, 295–305.Google Scholar
Nagasawa, M. (1961) The adjoint process of a diffusion with reflecting barrier. Kodai Math Sem. Rep. 13, 235–248.Google Scholar
Nagasawa, M. (1964) Time reversions of Markov processes. Nagoya Math. J. 24, 177–204.CrossRefGoogle Scholar
Nagasawa, M. (1975) Multiplicative Excessive Measures and Duality between Equations of Boltzmann and of Branching Processes. Lecture Notes in Mathematics 465, Springer-Verlag, Berlin, 471–485.Google Scholar
Nagasawa, M. (1977) Basic models of branching processes. Proc. 41st Session I.S.I., New Delhi. (2), 423–445.Google Scholar
Sawyer, S. (1977) On the past history of an allele now known to have frequency p. J. Appl. Prob. 14, 439–450.Google Scholar
Watterson, G. A. (1977) Reversibility and the age of an allele II: Two-allele models, with selection and mutation. Theoret. Popn Biol. 12, 179–196.Google Scholar
Weil, M. (1970) Quasi-processus. In Lecture Notes in Mathematics 124, Springer-Verlag, Berlin, 216–239.Google Scholar