Convergence in Wasserstein distance for empirical measures of Dirichlet diffusion processes on manifolds

Abstract

Let $M$ be a $d$-dimensional connected compact Riemannian manifold with boundary $\partial M$, let $V\in C^2(M)$ be such that $\mu (\mathrm{d}x):={\mathrm{e}}^{V(x)}\mathrm{d}x$ is a probability measure, and let $X_t$ be the diffusion process generated by $L:=\Delta +\nabla V$ with $\tau :=\inf \{t\ge 0:X_t\in \partial M\}$. Consider the empirical measure ${\mu }_t:=\frac{1}{t}{\int }_0^t{\delta }_{X_s}\mathrm{d}s$ under the condition $t<\tau$ for the diffusion process. If $d\le 3$, then for any initial distribution not fully supported on $\partial M$,

holds for some constant $c\in (0,1]$ with $c=1$ when $\partial M$ is convex, where ${\mu }_0:={\varphi }_0^2\mu$ for the first Dirichet eigenfunction ${\varphi }_0$ of $L$, $\{{\lambda }_m{\}}_{m\ge 0}$ are the Dirichlet eigenvalues of $-L$ listed in increasing order counting multiplicities, and the upper bound is finite if and only if $d\le 3$. When $d=4$, ${\sup }_{T\ge t}\mathbb{E}[{\mathbb{W}}_2({\mu }_t,{\mu }_0{)}^2|T<\tau ]$ decays on the order of $t^{-1}\log t$, while for $d\ge 5$ it behaves like $t^{-2/(d-2)}$, as $t\to \infty$.