Wasserstein Convergence Rate for Empirical Measures on Noncompact Manifolds

TL;DR

This paper investigates the rate at which empirical measures of reflecting diffusion processes on manifolds converge in Wasserstein distance, providing explicit bounds and sharp order results in specific geometric and potential settings.

Contribution

It derives explicit convergence rate bounds for empirical measures of diffusions on manifolds, including sharp order results in particular cases with unbounded potentials.

Findings

Explicit upper and lower bounds for convergence rates.

Sharp order results when dimension and potential parameters satisfy certain conditions.

Application to diffusions on Euclidean space with specific potentials.

Abstract

Let $X_t$ be the (reflecting) diffusion process generated by $L:=\Delta+\nabla V$ on a complete connected Riemannian manifold $M$ possibly with a boundary $\partial M$, where $V\in C^1(M)$ such that $\mu(d x):= e^{V(x)}d x$ is a probability measure. We estimate the convergence rate for the empirical measure $\mu_t:=\frac 1 t \int_0^t \delta_{X_s\d s$ under the Wasserstein distance. As a typical example, when $M=\mathbb R^d$ and $V(x)= c_1- c_2 |x|^p$ for some constants $c_1\in \mathbb R, c_2>0$ and $p>1$, the explicit upper and lower bounds are present for the convergence rate, which are of sharp order when either $d<\frac{4(p-1)}p$ or $d\ge 4$ and $p\to\infty$.