Asymptotic Formulas for Empirical Measures of (Reflecting) Diffusion Processes on Riemannian Manifolds

Feng-Yu Wang

Published 2019-06-08Version 1

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 taken for the diffusion process starting at point $x$, and $W_2$ is the $L^2$-Warsserstein distance induced by the Riemannian metric. The limit is finite if and only if $d\le 3$, and when $d\ge 4$ two-sided estimates are presented for the convergence rate of $\mathbb E[W_2(\mu_t,\mu)^2].$ The same type asymptotic formula is also derived for the modified empirical measures of the (reflecting) diffusion process on a class of non-compact Riemannian manifolds with or without boundary.