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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 

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