Convergence in Wasserstein distance for empirical measures of Dirichlet diffusion processes on manifolds

Let M be a d-dimensional connected compact Riemannian manifold with boundary ∂M, let V∈C2(M) such that μ(dx):=eV(x)dx is a probability measure, and let Xt be the diffusion process generated by L:=Δ+∇V with τ:=inf{t≥0:Xt∈∂M}. Consider the empirical measure μt:=1t∫t0δXsds under the condition t<τ for the diffusion process. If d≤3, then for any initial distribution not fully supported on ∂M,c∑m=1∞2(λm−λ0)2≤lim inft→∞infT≥t{tE[W2(μt,μ0)2∣∣T<τ]}≤lim supt→∞supT≥t{tE[W2(μt,μ0)2∣∣T<τ]}≤∑m=1∞2(λm−λ0)2holds for some constant c∈(0,1] with c=1 when ∂M is convex, where μ0:=ϕ20μ for the first Dirichet eigenfunction ϕ0 of L, {λm}m≥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≤3. When d=4, supT≥tE[W2(μt,μ0)2∣∣T