Sharp $L^q$-Convergence Rate in $p$-Wasserstein Distance for Empirical Measures of Diffusion Processes

TL;DR

This paper establishes precise rates of convergence in Wasserstein distance for empirical measures of diffusion processes, including non-symmetric cases, with explicit formulas involving eigenvalues and eigenfunctions.

Contribution

It derives the exact convergence rate for empirical measures of diffusion processes, including non-symmetric and reflecting diffusions, in Wasserstein distance, with explicit formulas for the limit behavior.

Findings

Convergence rate derived for empirical measures in Wasserstein distance.

Explicit limit formula involving eigenvalues and eigenfunctions.

Results hold uniformly for a range of p and q values.

Abstract

For a class of (non-symmetric) diffusion processes on a length space, which in particular include the (reflecting) diffusion processes on a connected compact Riemannian manifold, the exact convergence rate is derived for $({\mathbb E} [{\mathbb W}_p^q(\mu_T,\mu)])^{\frac{1}{q}} (T \to \infty)$ uniformly in $(p,q)\in [1,\infty) \times (0,\infty)$, where $\mu_T$ is the empirical measure of the diffusion process, $\mu$ is the unique invariant probability measure, and ${\mathbb W}_p$ is the $p$-Wasserstein distance. Moreover, when the dimension parameter is less than $4$, we prove that ${\mathbb E} |T {\mathbb W}_2^2(\mu_T,\mu)-\Xi(T)|^q \to 0$ as $T\to\infty$ for any $q\ge 1$, where $\Xi(T)$ is explicitly given by eigenvalues and eigenfunctions for the symmetric part of the generator.