Let M be a d-dimensional connected compact Riemannian manifold with boundary ∂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 $L:=\mathrm{\Delta }+\mathrm{\nabla }V$ with $\tau :=\inf \{t\ge 0:X_t\in \partial M\}$. Consider the conditional empirical measure ${\mu }_t^{\nu }:={\mathbb{E}}^{\nu }\left(\frac{1}{t}{\int }_0^t{\delta }_{X_s}\mathrm{d}s\right|t<\tau )$ for the diffusion process with initial distribution ν such that $\nu (\partial M)<1$. Then$\underset{t\to \infty }{\lim }{\{t{\mathbb{W}}_2({\mu }_t^{\nu },{\mu }_0)\}}^2=\frac{1}{{\{\mu ({\varphi }_0)\nu ({\varphi }_0)\}}^2}\sum _{m=1}^{\infty }\frac{{\{\nu ({\varphi }_0)\mu ({\varphi }_m)+\mu ({\varphi }_0)\nu ({\varphi }_m)\}}^2}{{({\lambda }_m-{\lambda }_0)}^3},$ where $\nu (f):={\int }_{M}f\mathrm{d}\nu$ for a measure ν and $f\in L^1(\nu )$, ${\mu }_0:={\varphi }_0^2\mu$, ${\{{\varphi }_m\}}_{m\ge 0}$ is the eigenbasis of −L in $L^2(\mu )$ with the Dirichlet boundary, ${\{{\lambda }_m\}}_{m\ge 0}$ are the corresponding Dirichlet eigenvalues, and ${\mathbb{W}}_2$ is the $L^2$-Wasserstein distance induced by the Riemannian metric.

让中号是d维连接的紧黎曼流形与边界∂中号，让$\mathrm{伏特}\in C^{\mathrm{2个}}（\mathrm{中号}）$ 这样 $\mu （\mathrm{d}X）：={\mathrm{Ë}}^{\mathrm{伏特}（X）}\mathrm{d}X$ 是一种概率测度， $X_{Ť}$ 是由产生的扩散过程 $\mathrm{大号}：=\mathrm{\Delta }+\mathrm{\nabla }\mathrm{伏特}$ 和 $\tau ：=\mathrm{信息}\{Ť\ge 0：X_{Ť}\in \partial \mathrm{中号}\}$。考虑有条件的经验测度${\mu }_{Ť}^{\nu }：={\mathbb{E}}^{\nu }（\frac{\mathrm{1个}}{Ť}{\int }_0^{Ť}{\delta }_{X_s}\mathrm{d}s|Ť<\tau ）$对于具有初始分布ν的扩散过程，使得$\nu （\partial \mathrm{中号}）<\mathrm{1个}$。然后$\underset{Ť\to \infty }{\mathrm{林}}{\{Ť{\mathbb{w\; ^}}_{\mathrm{2个}}（{\mu }_{Ť}^{\nu }，{\mu }_0）\}}^{\mathrm{2个}}=\frac{\mathrm{1个}}{{\{\mu （{\varphi }_0）\nu （{\varphi }_0）\}}^{\mathrm{2个}}}\sum _{米=\mathrm{1个}}^{\infty }\frac{{\{\nu （{\varphi }_0）\mu （{\varphi }_{米}）+\mu （{\varphi }_0）\nu （{\varphi }_{米}）\}}^{\mathrm{2个}}}{{（{\lambda }_{米}-{\lambda }_0）}^3}，$ 在哪里 $\nu （F）：={\int }_{\mathrm{中号}}F\mathrm{d}\nu$测度ν和$F\in {\mathrm{大号}}^{\mathrm{1个}}（\nu ）$， ${\mu }_0：={\varphi }_0^{\mathrm{2个}}\mu$， ${\{{\varphi }_{米}\}}_{米\ge 0}$是-L in的本征基${\mathrm{大号}}^{\mathrm{2个}}（\mu ）$ 与狄利克雷边界， ${\{{\lambda }_{米}\}}_{米\ge 0}$ 是对应的Dirichlet特征值，并且 ${\mathbb{w\; ^}}_{\mathrm{2个}}$ 是个 ${\mathrm{大号}}^{\mathrm{2个}}$黎曼度量引起的-Wasserstein距离。