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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 diffusionprocesses on $\DD_\infty:= \{{\bf x}\in [0,1]^\N: \sum_{i\ge 1} x_i=1\}$ withGEM distribution as the reversible measure. Log-Sobolev inequalities areestablished for these diffusions, which lead to the exponential convergence tothe corresponding reversible measures in the entropy. Extensions are made to aclass of measure-valued processes over an abstract space $S$. This provides areasonable alternative to the Fleming-Viot process which does not satisfy thelog-Sobolev inequality when $S$ is infinite as observed by W. Stannat \cite{S}.

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