A note on the $\mathcal{W}_2$-convergence rate of the empirical measure of an ergodic $\mathbb{R}^d$-valued diffusion

TL;DR

This paper investigates the convergence rate of the empirical measure of an ergodic diffusion process in Wasserstein distance, providing theoretical bounds under strong confluence and Lipschitz conditions.

Contribution

It offers new theoretical results on the $ ext{W}_2$-convergence rate of empirical measures for ergodic diffusions with specific regularity assumptions.

Findings

Established $ ext{W}_2$-convergence rates in mean quadratic sense.

Derived almost sure convergence rates.

Applicable to ergodic $ ext{R}^d$-valued diffusions under strong confluence.

Abstract

In this note, we consider a Stochastic Differential Equation under a strong confluence and Lipschitz continuity assumption of the coefficients. For the unique stationary solution, we study the rate of convergence of its empirical measure toward the invariant probability measure. We provide rate for the Wasserstein distance in the mean quadratic and almost sure sense.