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

Starting from a sequence of independent Wright-Fisher 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. Log-Sobolev 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 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 W. Stannat [13].