Limit Theorems in Warsserstein Distance for Empirical Measures of Diffusion Processes on Riemannian Manifolds (1906.03422v9)

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 $\mu$.