Precise Limit in Wasserstein Distance for Conditional Empirical Measures of Dirichlet Diffusion Processes

Let $M$ be a $d$-dimensional connected compact Riemannian manifold with boundary $\mathrm{\partial }M$, let $V\in C^2(M)$ such that $\mu (dx):=e^{V(x)}dx$ is a probability measure, and let $X_t$ be the diffusion process generated by $L:=\mathrm{\Delta }+\mathrm{\nabla }V$ with $\tau :=\inf \{t\ge 0:X_t\in \mathrm{\partial }M\}$. Consider the conditional empirical measure $\mu_t^\nu:= \mathbb E^\nu\big(\frac 1 t \int_0^t \delta_{X_s}d s\big|t<\tau\big)$forthediffusionprocesswithinitialdistribution$\nu$suchthat$\nu(\partial M)<1$\mathrm{.}Then$$\lim_{t\to\infty} \big{t\mathbb W_2(\mu_t^\nu,\mu_0)\big}^2 = \frac 1 {{\mu(\phi_0)\nu(\phi_0)}^2} \sum_{m=1}^\infty \frac{{\nu(\phi_0)\mu(\phi_m)+ \mu(\phi_0) \nu(\phi_m)}^2}{(\lambda_m-\lambda_0)^3},$$where$\nu(f):=\int_Mf {d} \nu$forameasure$\nu$and$f\in L^1(\nu)$,$\mu_0:=\phi_0^2\mu$,${\phi_m}_{m\ge 0}$istheeigenbasisof$-L$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.