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

Abstract

Let $M$ be a compact connected Riemannian manifold possibly with a boundary, let $V\in C^2(M)$ such that $\mu (dx):=e^{V(x)}dx$ 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 $$\underset{t\to \infty }{\lim }\{t{\mathbb{E}}^x[W_2({\mu }_t,\mu {)}^2]\}=\sum _{i=1}^{\infty }\frac{2}{{\lambda }_i^2}\text{ uniformly in }x\in M,$$ 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: $$\underset{t\to \infty }{\lim }\underset{x\in M}{\sup }|{\mathbb{P}}^x(t{W}_2({\mu }_t,\mu {)}^2<a)-\mathbb{P}(\sum _{k=1}^{\infty }\frac{2{\xi }_k^2}{{\lambda }_k^2}<a)|=0,a\ge 0,$$ 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 $\mu$.

48 pages