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) d x is a probability measure, and let Xt be the diffusion process generated by L:=+∇ V with τ:=∈f\t 0: Xt∈∂ M\. Consider the conditional empirical measure μt:= E( 1 t ∫0t δXsd s|t<τ) for the diffusion process with initial distribution such that (∂ M)<1. Then t∞ \t W2(μt,μ0)\2 = 1 \μ(φ0)(φ0)\2 Σm=1∞ \(φ0)μ(φm)+ μ(φ0) (φm)\2(λm-λ0)3, where (f):=∫Mf d for a measure and f∈ L1(), 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.