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 $DD_infty:= {{bf x}in [0,1]^N: sum_{ige 1} x_i=1}$ 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 cite{S}.