Precise limit in Wasserstein distance for conditional empirical measures of Dirichlet diffusion processes

Abstract

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 conditional empirical measure μtν:=Eν(1t∫0tδXsds|t<τ) for the diffusion process with initial distribution ν such that 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 for a measure ν and f∈L1(ν), μ0:=ϕ02μ, {ϕm}m≥0 is the eigenbasis of −L in L2(μ) with the Dirichlet boundary, {λm}m≥0 are the corresponding Dirichlet eigenvalues, and W2 is the L2-Wasserstein distance induced by the Riemannian metric.