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 $V\in C^2(M)$ 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 $\partial M$, \begin{align*} &c\sum_{m=1}^\infty \frac{2}{(λ_m-λ_0)^2} \le \liminf_{t\to \infty} \inf_{T\ge t} \Big\{t {\mathbb E}\big[\mathbb W_2(μ_t, μ_0)^2\big|T<τ\big]\Big\} \\ &\le \limsup_{t\to \infty} \sup_{T\ge t} \Big\{ t \mathbb E\big[\mathbb W_2(μ_t, μ_0)^2\big|T<τ\big] \Big\}\le \sum_{m=1}^\infty \frac{2}{(λ_m-λ_0)^2}\end{align*} 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 the 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 in the order $t^{-1}\log t$, while for $d\ge 5$ it behaves like $t^{-\frac{2}{d-2}}$, as $t\to \infty$.

27 pages

References in corpus (1)

• Formulae for the derivatives of heat semigroups

Cited by in corpus (2)

• Wasserstein Convergence Rate for Empirical Measures on Noncompact Manifolds
• Precise Limit in Wasserstein Distance for Conditional Empirical Measures of Dirichlet Diffusion Processes