A New Probability Measure-Valued Stochastic Process with Ferguson-Dirichlet Process as Reversible Measure

Abstract

A new diffusion process taking values in the space of all probability measures over $[0,1]$ is constructed through Dirichlet form theory in this paper. This process is reversible with respect to the Ferguson-Dirichlet process (also called Poisson Dirichlet process), which is the reversible measure of the Fleming-Viot process with parent independent mutation. The intrinsic distance of this process is in the class of Wasserstein distances, so it's also a kind of Wasserstein diffusion. Moreover, this process satisfies the Log-Sobolev inequality.