Wasserstein convergence rate for empirical measures on noncompact manifolds

Feng-Yu Wang

Stochastic Processes and their Applications, 2022, vol. 144, issue C, 271-287

Abstract: Let Xt be the (reflecting) diffusion process generated by L≔Δ+∇V on a complete connected Riemannian manifold M possibly with a boundary ∂M, where V∈C1(M) such that μ(dx)≔eV(x)dx is a probability measure. We estimate the convergence rate for the empirical measure μt≔1t∫0tδXsds under the Wasserstein distance. As a typical example, when M=Rd and V(x)=c1−c2|x|p for some constants c1∈R,c2>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→∞.

Keywords: Empirical measure; Diffusion process; Wasserstein distance; Riemannian manifold (search for similar items in EconPapers)
Date: 2022
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Citations: View citations in EconPapers (2)

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DOI: 10.1016/j.spa.2021.11.006

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