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

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

2007-11-12

arxiv.org/abs/0711.1887v1

Mathematics

Probability

14 pages

Scientific paper

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}.

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