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

TL;DR

This note establishes quantitative rates for the Wasserstein convergence of the empirical measure of an ergodic diffusion in $\mathbb{R}^d$ to its invariant distribution under Lipschitz and strong confluence conditions. The authors combine smoothing techniques with a coboundary/Poisson-equation framework to obtain $L^2$ rates and almost-sure rates, showing polynomial and exponential-moment dependent decay and highlighting improvements when the diffusion coefficient is bounded or uniformly elliptic. A key contribution is the coboundary decomposition of time averages, enabling almost-sure rates that extend beyond previous results for self-interacting diffusions. The results have practical implications for ergodic simulation and numerical methods targeting stationary distributions in diffusion and SID/McKean-Vlasov-type systems, clarifying how moment and ellipticity conditions govern convergence speeds.

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.

Theorems & Definitions (14)

• Lemma 1.1
• Theorem 1.1
• Theorem 1.2
• Remark 1.1
• Proposition 2.1
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 3.1
• Proposition 3.1