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Wasserstein convergence rate for empirical measures on noncompact manifolds

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  • Wang, Feng-Yu

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→∞.

Suggested Citation

  • Wang, Feng-Yu, 2022. "Wasserstein convergence rate for empirical measures on noncompact manifolds," Stochastic Processes and their Applications, Elsevier, vol. 144(C), pages 271-287.
    • Handle: RePEc:eee:spapps:v:144:y:2022:i:c:p:271-287
    DOI: 10.1016/j.spa.2021.11.006
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    References listed on IDEAS

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    1. Arnaudon, Marc & Thalmaier, Anton & Wang, Feng-Yu, 2009. "Gradient estimates and Harnack inequalities on non-compact Riemannian manifolds," Stochastic Processes and their Applications, Elsevier, vol. 119(10), pages 3653-3670, October.
    2. Пигнастый, Олег & Koжевников, Георгий, 2019. "Распределенная Динамическая Pde-Модель Программного Управления Загрузкой Технологического Оборудования Производственной Линии [Distributed dynamic PDE-model of a program control by utilization of t," MPRA Paper 93278, University Library of Munich, Germany, revised 02 Feb 2019.
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    Cited by:

    1. Huaiqian Li & Bingyao Wu, 2023. "Wasserstein Convergence Rates for Empirical Measures of Subordinated Processes on Noncompact Manifolds," Journal of Theoretical Probability, Springer, vol. 36(2), pages 1243-1268, June.
    2. Huesmann, Martin & Mattesini, Francesco & Trevisan, Dario, 2023. "Wasserstein asymptotics for the empirical measure of fractional Brownian motion on a flat torus," Stochastic Processes and their Applications, Elsevier, vol. 155(C), pages 1-26.

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