Convergence in Wasserstein Distance for Empirical Measures of Dirichlet Diffusion Processes on Manifolds

TL;DR

This paper investigates the convergence rates of empirical measures of Dirichlet diffusion processes on compact manifolds in Wasserstein distance, revealing dimension-dependent decay behaviors and spectral bounds.

Contribution

It provides sharp bounds for Wasserstein convergence of empirical measures on manifolds, extending understanding of diffusion process behavior in various dimensions.

Findings

Upper bounds on Wasserstein distance decay depend on manifold dimension.

Explicit decay rates are derived for dimensions 4 and higher.

Convergence bounds relate to Dirichlet eigenvalues of the generator.

Abstract

Let $M$ be a $d$-dimensional connected compact Riemannian manifold with boundary $\partial M$, let $V\in C^2(M)$ such that $\mu({\rm d} x):={\rm e}^{V(x)}{\rm d} x$ is a probability measure, and let $X_t$ be the diffusion process generated by $L:=\Delta+\nabla V$ with $\tau:=\inf\{t\ge 0: X_t\in\partial M\}$. Consider the empirical measure $\mu_t:=\frac 1 t \int_0^t \delta_{X_s}{\rm d} s$ under the condition $t<\tau$ for the diffusion process. If $d\le 3$, then for any initial distribution not fully supported on $\partial M$, \begin{align*} &c\sum_{m=1}^\infty \frac{2}{(\lambda_m-\lambda_0)^2} \le \liminf_{t\to \infty} \inf_{T\ge t} \Big\{t {\mathbb E}\big[\mathbb W_2(\mu_t, \mu_0)^2\big|T<\tau\big]\Big\} \\ &\le \limsup_{t\to \infty} \sup_{T\ge t} \Big\{ t \mathbb E\big[\mathbb W_2(\mu_t, \mu_0)^2\big|T<\tau\big] \Big\}\le \sum_{m=1}^\infty \frac{2}{(\lambda_m-\lambda_0)^2}\end{align*} holds for some constant $c\in (0,1]$ with $c=1$ when $\partial M$ is convex, where $\mu_0:= \phi_0^2\mu$ for the first Dirichet eigenfunction $\phi_0$ of $L$, $\{\lambda_m\}_{m\ge 0}$ are the Dirichlet eigenvalues of $-L$ listed in the increasing order counting multiplicities, and the upper bound is finite if and only if $d\le 3$. When $d=4$, $\sup_{T\ge t} \mathbb E\big[\mathbb W_2(\mu_t, \mu_0)^2\big|T<\tau\big] $ decays in the order $t^{-1}\log t$, while for $d\ge 5$ it behaves like $t^{-\frac 2 {d-2}}$, as $t\to\infty$.