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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 $\partial M$, let $V\in C^2(M)$ such that $μ(dx):=e^{V(x)} d x$ is a probability measure, and let $X_t$ be the diffusion process generated by $L:=Δ+\nabla V$ with $τ:=\inf\{t\ge 0: X_t\in\partial M\}$. Consider the conditional empirical measure $μ_t^ν:= \mathbb E^ν\big(\frac 1 t \int_0^t δ_{X_s}d s\big|t<τ\big)$ for the diffusion process with initial distribution $ν$ such that $ν(\partial M)<1$. Then $$\lim_{t\to\infty} \big\{t\mathbb W_2(μ_t^ν,μ_0)\big\}^2 = \frac 1 {\{μ(ϕ_0)ν(ϕ_0)\}^2} \sum_{m=1}^\infty \frac{\{ν(ϕ_0)μ(ϕ_m)+ μ(ϕ_0) ν(ϕ_m)\}^2}{(λ_m-λ_0)^3},$$ where $ν(f):=\int_Mf {d} ν$ for a measure $ν$ and $f\in L^1(ν)$, $μ_0:=ϕ_0^2μ$, $\{ϕ_m\}_{m\ge 0}$ is the eigenbasis of $-L$ in $L^2(μ)$ with the Dirichlet boundary, $\{λ_m\}_{m\ge 0}$ are the corresponding Dirichlet eigenvalues, and $\mathbb W_2$ is the $L^2$-Wasserstein distance induced by the Riemannian metric.