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

Jinghai Shao

doi:10.1214/ejp.v16-844

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.

Highlights

This work is motivated by Von Renesse, M-K. and Sturm, K.T. [24] about Wasserstein diffusion on one dimensional space
There they constructed a probability measure-valued stochastic process, which is reversible with respect to an “entropy measure"
As proved in [7], it is the reversible measure of the Fleming-Viot processes with parent independent mutation

Summary

Introduction

This work is motivated by Von Renesse, M-K. and Sturm, K.T. [24] about Wasserstein diffusion on one dimensional space. As proved in [7], it is the reversible measure of the Fleming-Viot processes with parent independent mutation It was shown in [7] and [8] that under the Ferguson-Dirichlet process, almost surely the measure μ ∈ ([0, 1]) is discrete and has full topological support. Based on the works [24] and [3], it’s of great interest to ask whether there exists a probability measure-valued stochastic process, which is reversible w.r.t. Ferguson-Dirichlet process and satisfies the Log-Sobolev inequality at the same time. The last section is devoted to establish the Log-Sobolev inequality for our new probability measure-valued process This is proved by the method of finite dimensional approximation based on our construction of Dirichlet form and Döring, M. and Stannat, W.’s work [3]

Comparison with Fleming-Viot process

Quasi-invariance property

Integration by parts formula

Tangent space and Dirichlet form

Log-Sobolev inequalities for the process