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A class of infinite dimensional diffusion processes with connection to population genetics
 Journal of Applied Probability
, 2007
"... Starting from a sequence of independent WrightFisher diffusion processes on [0, 1], we construct a class of reversible infinite dimensional diffusion processes on ∆ ∞: = {x ∈ [0, 1] N: i≥1 xi = 1} with GEM distribution as the reversible measure. LogSobolev inequalities are established for these di ..."
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Starting from a sequence of independent WrightFisher diffusion processes on [0, 1], we construct a class of reversible infinite dimensional diffusion processes on ∆ ∞: = {x ∈ [0, 1] N: i≥1 xi = 1} with GEM distribution as the reversible measure. LogSobolev inequalities are established for these diffusions, which lead to the exponential convergence to the corresponding reversible measures in the entropy. Extensions are made to a class of measurevalued processes over an abstract space S. This provides a reasonable alternative to the FlemingViot process which does not satisfy the logSobolev inequality when S is infinite as observed by W. Stannat [13].
A new probability measurevalued stochastic process with FergusonDirichlet process as reversible measure ∗
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TITLE: Asymptotic Theory for Three Infinite Dimensional Diffusion Processes
"... ii This thesis is centered around three infinite dimensional diffusion processes: (i). the infinitelymanyneutralalleles diffusion model [Ethier and Kurtz, 1981], (ii). the twoparameter infinite dimensional diffusion model [Petrov, 2009] and [Feng and Sun, 2010], (iii). the infinitelymanyallele ..."
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ii This thesis is centered around three infinite dimensional diffusion processes: (i). the infinitelymanyneutralalleles diffusion model [Ethier and Kurtz, 1981], (ii). the twoparameter infinite dimensional diffusion model [Petrov, 2009] and [Feng and Sun, 2010], (iii). the infinitelymanyalleles diffusion with symmetric dominance [Ethier and Kurtz, 1998]. The partition structures, the ergodic inequalities and the asymptotic theory of these three models are discussed. In particular, the asymptotic theory turns out to be the major contribution of this thesis. In Chapter 2, a slightly altered version of Kingman’s onetoone correspondence theorem on partition structures is provided, which in turn becomes a handy tool for obtaining the asymptotic result on the partition structures associated with models (i) and (ii).