A Monotone Approximation to the Wasserstein Diffusion

TL;DR

This paper introduces a finite-dimensional approximation of the Wasserstein diffusion, a Markov process on probability measures, using interacting Brownian motions to facilitate analysis and computation.

Contribution

It develops a novel approximation scheme for the Wasserstein diffusion via finite-dimensional interacting Brownian systems, bridging infinite and finite-dimensional stochastic processes.

Findings

Successfully approximates Wasserstein diffusion with finite systems

Provides a framework for analyzing infinite-dimensional diffusions

Enhances computational approaches for Wasserstein space processes

Abstract

Von Renesse and the author (Ann. Prob. '09) developed a second order calculus on the Wasserstein space P([0,1]) of probability measures on the unit interval. The basic objects of interest had been Dirichlet form, semigroup and continuous Markov process, called Wasserstein diffusion. The goal of this paper is to derive approximations of these objects on the infinite dimensional space P([0, 1]) in terms of appropriate objects on finite dimensional spaces. In particular, we will approximate the Wasserstein diffusion in terms of interacting systems of Brownian motions.