A Class of Infinite Dimensional Diffusion Processes with Connection to Population Genetics
Abstract
Starting from a sequence of independent Wright-Fisher diffusion processes on
$[0,1]$, we construct a class of reversible infinite dimensional diffusion
processes on $\DD_\infty:= \{{\bf x}\in [0,1]^\N: \sum_{i\ge 1} x_i=1\}$ 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}.