Entropic Measure and Wasserstein Diffusion

TL;DR

This paper introduces new Gibbs-structured random probability measures on the sphere and interval, and constructs diffusion processes on these measures using Dirichlet form methods, including a Wasserstein space diffusion.

Contribution

It develops a novel Gibbs measure framework with entropy-based Hamiltonian and constructs canonical Wasserstein space diffusion processes.

Findings

Constructed Gibbs measures with quasi-invariance properties.

Developed diffusion processes related to Malliavin's Brownian motion.

Established a canonical Wasserstein space diffusion process.

Abstract

We construct a new random probability measure on the sphere and on the unit interval which in both cases has a Gibbs structure with the relative entropy functional as Hamiltonian. It satisfies a quasi-invariance formula with respect to the action of smooth diffeomorphism of the sphere and the interval respectively. The associated integration by parts formula is used to construct two classes of diffusion processes on probability measures (on the sphere or the unit interval) by Dirichlet form methods. The first one is closely related to Malliavin's Brownian motion on the homeomorphism group. The second one is a probability valued stochastic perturbation of the heat flow, whose intrinsic metric is the quadratic Wasserstein distance. It may be regarded as the canonical diffusion process on the Wasserstein space.