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The geometry of random drift IV. Random time substitutions and stationary densities

Published online by Cambridge University Press:  01 July 2016

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

In the present paper a spatially homogeneous distance measure of Edwards and Cavalli-Sforza type is derived for multiple allele random genetic drift with a (possibly vanishing) symmetric mutation field, using the technique of random time substitution. Since the mutation field is of gradient type in gene frequency space, the transformed process is proved to be Brownian motion relative to a new Riemannian geometry and a new time measure. The new geometry is conformally related to spherical geometry of the original process but is not of constant curvature, generally. A formula relating the stationary density of the old and new process is derived and the edge length formula for the new geometry on the n-simplex frequency space is given and analysed.

Keywords

GENETIC DISTANCEWFK-PROCESS DIFFUSIONRANDOM TIME SUBSTITUTIONTRANSFORMED DIFFUSIONCONFORMAL DIFFERENTIAL GEOMETRYCURVATURESTATIONARY DENSITIESBROWNIAN MOTION IN RIEMANNIAN GEOMETRY
Type
Research Article
Information
Advances in Applied Probability , Volume 10 , Issue 3 , September 1978 , pp. 563 - 569
Copyright
Copyright © Applied Probability Trust 1978 

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References

Antonelli, P. L. (1978) An asymptotic formula for the transition density of random genetic drift. J. Appl. Prob. 15, 185–187.Google Scholar
Antonelli, P. L., Chapin, J., Lathrop, G. M. and Morgan, K. (1977) The geometry of random drift II. The symmetry of random genetic drift. Adv. Appl. Prob. 9, 250–259.Google Scholar
Antonelli, P. L. and Morgan, K. (1978) On the persistence of random genetic drift. J. Appl. Prob. 15, 180–184.Google Scholar
Antonelli, P. L., Morgan, K. and Lathrop, G. M. (1977) The geometry of random drift II. Recombination and diffusion. Adv. Appl. Prob. 9, 260–267.Google Scholar
Antonelli, P. L. and Strobeck, C. (1977) The geometry of random drift I. Stochastic distance and diffusion. Adv. Appl. Prob. 9, 238–249.Google Scholar
Dynkin, E. B. (1965) Markov Processes, Vol. I. Academic Press, New York.Google Scholar
Edwards, A. W. F. and Cavalli-Sforza, L. L. (1972) Affinity as revealed by differences in gene frequencies. In The Assessment of Population Affinities in Man, ed. Weiner, J. and Huizinga, J.. Clarendon Press, Oxford.Google Scholar
Eisenhart, L. P. (1926) Riemannian Geometry. Princeton University Press, Princeton, N.J. Google Scholar
Korn, G. A. and Korn, T. M. (1968) Mathematical Handbook for Scientists and Engineers, 2nd. edn. McGraw-Hill, New York.Google Scholar