  Let $M$ be a $d$-dimensional connected compact Riemannian manifold with
boundary $\partial M$, let $V\in C^2(M)$ such that $\mu(dx):=e^{V(x)} d x$ 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 conditional empirical measure
  $\mu_t^\nu:= \mathbb E^\nu\big(\frac 1 t \int_0^t \delta_{X_s}d
s\big|t<\tau\big)$ for the diffusion process with initial distribution $\nu$
such that $\nu(\partial M)<1$. 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 $\nu(f):=\int_Mf {d} \nu$
for a measure $\nu$ and $f\in L^1(\nu)$, $\mu_0:=\phi_0^2\mu$,
$\{\phi_m\}_{m\ge 0}$ 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.
