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

摘要

Let Mbe a d-dimensional connected compact Riemannian manifold with boundary. M, let V is an element of C-2(M) such that mu(dx) := e(V)(x)dxis a probability measure, and let Xtbe the diffusion process generated by L := Delta + del V with Y:= inf{t >= 0 : X-t is an element of partial derivative M}. Consider the conditional empirical measure mu(nu)(t) = E-nu (1/t integral(t delta)(0)X(s)ds vertical bar t infinity) {tW2((mu(nu,)(t) mu 0)}(2) =1/{mu(phi 0)nu(phi 0)}(2) Sigma(infinity)(m=1) {nu(phi(0))mu(phi(m)) + mu(phi(0))nu(phi(m))}(2)/(lambda(m) - lambda(0))(3) where.(f) := Mfd.for a measure.and f. L1(nu), mu(0):= phi 20 mu, {phi m} m=0 is the eigenbasis of -Lin L2(mu) with the Dirichlet boundary, {.m} m=0are the corresponding Dirichlet eigenvalues, and W-2 is th L-2-Wasserstein distance induced by the Riemannian metric. (C) 2021 Elsevier Inc. All rights reserved.