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 ∂M, let V∈C2(M) such that μ(dx):=eV(x)dx is probability measure, and Xt the diffusion process generated by L:=Δ+∇V τ:=inf⁡{t≥0:Xt∈∂M}. Consider conditional empirical measure μtν:=Eν(1t∫0tδXsds|t<τ) for initial distribution ν ν(∂M)<1. Thenlimt→∞⁡{tW2(μtν,μ0)}2=1{μ(ϕ0)ν(ϕ0)}2∑m=1∞{ν(ϕ0)μ(ϕm)+μ(ϕ0)ν(ϕm)}2(λm−λ0)3, where ν(f):=∫Mfdν f∈L1(ν), μ0:=ϕ02μ, {ϕm}m≥0 eigenbasis of −L in L2(μ) Dirichlet boundary, {λm}m≥0 are corresponding eigenvalues, W2 L2-Wasserstein distance induced metric.