  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 $\mu$.
