Asymptotic behavior of Wasserstein distance for weighted empirical measures of diffusion processes on compact Riemannian manifolds

TL;DR

This paper studies the long-term behavior of weighted empirical measures of diffusion processes on compact Riemannian manifolds, focusing on the Wasserstein distance convergence and providing sharper limit theorems for specific weights.

Contribution

It establishes the asymptotic behavior of the expected Wasserstein distance between weighted empirical measures and the invariant measure, refining previous results for the case when the weight parameter equals one.

Findings

Asymptotic formulas for Wasserstein distance expectations

Sharper limit theorem for the case α=1

Method combines PDE and mass transportation techniques

Abstract

Let $(X_t)_{t \geq 0}$ be a diffusion process defined on a compact Riemannian manifold, and for $\alpha > 0$, let $$ \mu_t^{(\alpha)} = \frac{\alpha}{t^\alpha} \int_{0}^{t} \delta_{X_s} \, s^{\alpha - 1} \mathrm{d} s $$ be the associated weighted empirical measure. We investigate asymptotic behavior of $\mathbb{E}^\nu \big[ \mathrm{W}_2^2(\mu_t^{(\alpha)}, \mu) \big]$ for sufficient large $t$, where $\mathrm{W}_2$ is the quadratic Wasserstein distance and $\mu$ is the invariant measure of the process. In the particular case $\alpha = 1$, our result sharpens the limit theorem achieved in [26]. The proof is based on the PDE and mass transportation approach developed by L. Ambrosio, F. Stra and D. Trevisan.