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

TL;DR

This paper constructs a new class of infinite-dimensional diffusion processes linked to population genetics, establishing their properties and convergence behavior, and offering an alternative to the Fleming-Viot process in infinite spaces.

Contribution

It introduces a novel class of reversible infinite-dimensional diffusions with GEM distribution, extending to measure-valued processes and establishing log-Sobolev inequalities.

Findings

Proves exponential convergence in entropy to the reversible measure.

Establishes log-Sobolev inequalities for the new diffusion class.

Provides an alternative to Fleming-Viot processes in infinite spaces.

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}.