Limit Theorems in Warsserstein Distance for Empirical Measures of Diffusion Processes on Riemannian Manifolds

TL;DR

This paper establishes limit theorems for the Wasserstein distance between empirical measures of diffusion processes on Riemannian manifolds, revealing different asymptotic behaviors depending on the dimension and providing a large deviation principle.

Contribution

It provides new limit theorems, including a central limit theorem and asymptotic order results, for empirical measures of diffusion processes on manifolds, extending understanding of Wasserstein distances.

Findings

Finite limit for d ≤ 3, with a CLT for the Wasserstein distance squared.

Asymptotic order of expectation for d ≥ 4 is t^{-2/(d-2)}.

Long-time large deviation principle with a rate function based on relative entropy.

Abstract

Let $M$ be a compact connected Riemannian manifold possibly with a boundary, let $V\in C^2(M)$ such that $\mu(d x):=e^{V(x)}d x$ is a probability measure, and let $\{\lambda_i\}_{i\ge 1} $ be all non-trivial eigenvalues of $-L$ with Neumann boundary condition if the boundary exists. Then the empirical measures $\{\mu_t\}_{t>0}$ of the diffusion process generated by $L$ (with reflecting boundary if the boundary exists) satisfy $$ \lim_{t\to \infty} \big\{t \mathbb E^x [W_2(\mu_{t},\mu)^2]\big\}= \sum_{i=1}^\infty\frac 2 {\lambda_i^2}\ \text{ uniformly\ in\ } x\in M,$$ where $\mathbb E^x$ denotes the expectation for the diffusion process starting at point $x$, $W_2$ is the $L^2$-Warsserstein distance induced by the Riemannian metric. The limit is finite if and only if $d\le 3$, and in this case we derive the following central limit theorem: $$\lim_{t\to\infty} \sup_{x\in M} \Big|\mathbb P^x(t W_2(\mu_{t},\mu)^2<a)- \mathbb P\Big( \sum_{k=1}^\infty \frac{2\xi_k^2}{\lambda_k^2}<a\Big)\Big|=0, \ \ a\ge 0,$$ where $\mathbb P^x$ is the probability with respect to $\mathbb E^x$, and $\{\xi_k\}_{k\ge 1}$ are i.i.d. standard Gaussian random variables. Moreover, when $d\ge 4$ we prove that the main order of $\mathbb E^x[W_2(\mu_{t},\mu)^2]$ is $t^{-\frac 2 {d-2}}$ as $t\to\infty$. Moreover, when $d\ge 4$ the main order of $\mathbb E^x[W_2(\mu_{t},\mu)^2]$ is $t^{-\frac 2 {d-2}}$ as $t\to\infty$. Finally, we establish the long-time large deviation principle for $\{W_2(\mu_t,\mu)^2\}_{t\ge 0}$ with a good rate function given by the information with respect to $\mu$.