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A Class of Infinite-Dimensional Diffusion Processes with Connection to Population Genetics

Part of: Limit theorems

Published online by Cambridge University Press:  14 July 2016

Shui Feng  and
Feng-Yu Wang
Shui Feng*
Affiliation:
McMaster University
Feng-Yu Wang*
Affiliation:
Beijing Normal University and Swansea University
*
∗Postal address: Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada L8S 4K1. Email address: shuifeng@mcmaster.ca
∗∗Postal address: Department of Mathematics, Swansea University, Singleton Park, Swansea SA2 8PP, UK. Email address: f.y.wang@swansea.ac.uk
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Abstract

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Starting from a sequence of independent Wright-Fisher diffusion processes on [0, 1], we construct a class of reversible infinite-dimensional diffusion processes on Δ∞ := {x ∈ [0, 1]N: ∑i≥1xi = 1} with GEM distribution as the reversible measure. Log-Sobolev inequalities are established for these diffusions, which lead to the exponential convergence of the corresponding reversible measures in the entropy. Extensions are made to a class of measure-valued processes over an abstract space S. This provides a reasonable alternative to the Fleming-Viot process, which does not satisfy the log-Sobolev inequality when S is infinite as observed by Stannat (2000).

Keywords

Poisson-Dirichlet distributionGEM distributionFleming-Viot processlog-Sobolev inequality

MSC classification

Secondary: 60F10: Large deviations 92D10: Genetics
Type
Research Papers
Information
Journal of Applied Probability , Volume 44 , Issue 4 , December 2007 , pp. 938 - 949
Copyright
Copyright © Applied Probability Trust 2007 

References

[1] Bakry, D. (1997). On Sobolev and logarithmic Sobolev inequalities for Markov semigroups. In New Trends in Stochastic Analysis, eds Elworthy, K. D. et al., World Scientific, Singapore, pp. 43–75.Google Scholar
[2] Ethier, S. N. and Kurtz, T. G. (1981). The infinitely-many-neutral-alleles diffusion model. Adv. Appl. Prob. 13, 429–452.CrossRefGoogle Scholar
[3] Feng, S. and Hoppe, F. M. (1998). Large deviation principles for some random combinatorial structures in population genetics and Brownian motion. Ann. Appl. Prob. 8, 975–994.CrossRefGoogle Scholar
[4] Fukushima, M., Oshima, Y. and Takeda, M. (1994). Dirichlet Forms and Symmetric Markov Processes. De Gruyter, Berlin.CrossRefGoogle Scholar
[5] Gross, L. (1976). Logarithmic Sobolev inequalities. Amer. J. Math. 97, 1061–1083.CrossRefGoogle Scholar
[6] Gross, L. (1993). Logarithmic Sobolev Inequalities and Contractivity Properties of Semigroups (Lecture Notes Math. 1563). Springer, Berlin.CrossRefGoogle Scholar
[7] Hoppe, F. M. (1984). Pólya-like urns and the Ewens sampling formula. J. Math. Biol. 20, 91–94.CrossRefGoogle Scholar
[8] Kingman, J. F. C. et al. (1975). Random discrete distribution. J. R. Statist. Soc. Ser. B 37, 1–22.Google Scholar
[9] Ma, Z. M. and Röckner, M. (1992). Introduction to the Theory of (Nonsymmetric) Dirichlet Forms. Springer, Berlin.CrossRefGoogle Scholar
[10] Pitman, J. (1995). Exchangeable and partially exchangeable random partitions. Prob. Theory Relat. Fields 102, 145–158.CrossRefGoogle Scholar
[11] Pitman, J. (1996). Random discrete distributions invariant under size-biased permutation. Adv. Appl. Prob. 28, 525–539.CrossRefGoogle Scholar
[12] Pitman, J. and Yor, M. (1997). The two-parameter Poisson–Dirichlet distribution derived from a stable subordinator. Ann. Prob. 25, 855–900.CrossRefGoogle Scholar
[13] Stannat, W. (2000). On the validity of the log-Sobolev inequality for symmetric Fleming–Viot operators. Ann. Prob. 28, 667–684.CrossRefGoogle Scholar