Wasserstein convergence rate for empirical measures on noncompact manifolds

Abstract Let X t be the (reflecting) diffusion process generated by L ≔ Δ + ∇ V on a complete connected Riemannian manifold M possibly with a boundary ∂ M , where V ∈ C 1 ( M ) such that μ ( d x ) ≔ e V ( x ) d x is a probability measure. We estimate the convergence rate for the empirical measure μ t ≔ 1 t ∫ 0 t δ X s d s under the Wasserstein distance. As a typical example, when M = R d and V ( x ) = c 1 − c 2 | x | p for some constants c 1 ∈ R , c 2 > 0 and p > 1 , the explicit upper and lower bounds are present for the convergence rate, which are of sharp order when either d 4 ( p − 1 ) p or d ≥ 4 and p → ∞ .