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Convergence in Wasserstein distance for empirical measures of Dirichlet diffusion processes on manifolds

Feng-yu Wang

Journal of the European Mathematical Society, Volume: 25, Issue: 9

Swansea University Author: Feng-yu Wang

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DOI (Published version): 10.4171/jems/1269

Abstract

Let M be a d-dimensional connected compact Riemannian manifold with boundary ∂M, let V∈C2(M) such that μ(dx):=eV(x)dx is a probability measure, and let Xt be the diffusion process generated by L:=Δ+∇V with τ:=inf{t≥0:Xt∈∂M}. Consider the empirical measure μt:=1t∫t0δXsds under the condition t<τ fo...

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Published in: Journal of the European Mathematical Society
ISSN: 1435-9855 1435-9863
Published: European Mathematical Society - EMS - Publishing House GmbH 2022
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URI: https://cronfa.swan.ac.uk/Record/cronfa60533
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last_indexed 2025-05-16T08:16:37Z
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spelling 2025-05-15T16:04:15.8056318 v2 60533 2022-07-19 Convergence in Wasserstein distance for empirical measures of Dirichlet diffusion processes on manifolds 6734caa6d9a388bd3bd8eb0a1131d0de Feng-yu Wang Feng-yu Wang true false 2022-07-19 Let M be a d-dimensional connected compact Riemannian manifold with boundary ∂M, let V∈C2(M) such that μ(dx):=eV(x)dx is a probability measure, and let Xt be the diffusion process generated by L:=Δ+∇V with τ:=inf{t≥0:Xt∈∂M}. Consider the empirical measure μt:=1t∫t0δXsds under the condition t<τ for the diffusion process. If d≤3, then for any initial distribution not fully supported on ∂M,c∑m=1∞2(λm−λ0)2≤lim inft→∞infT≥t{tE[W2(μt,μ0)2∣∣T<τ]}≤lim supt→∞supT≥t{tE[W2(μt,μ0)2∣∣T<τ]}≤∑m=1∞2(λm−λ0)2holds for some constant c∈(0,1] with c=1 when ∂M is convex, where μ0:=ϕ20μ for the first Dirichet eigenfunction ϕ0 of L, {λm}m≥0 are the Dirichlet eigenvalues of −L listed in the increasing order counting multiplicities, and the upper bound is finite if and only if d≤3. When d=4, supT≥tE[W2(μt,μ0)2∣∣T<τ] decays in the order t−1logt, while for d≥5 it behaves like t−2d−2, as t→∞. Journal Article Journal of the European Mathematical Society 25 9 European Mathematical Society - EMS - Publishing House GmbH 1435-9855 1435-9863 Conditional empirical measure, Dirichlet diffusion process, Wasserstein distance, eigenvalues, eigenfunctions. 3 9 2022 2022-09-03 10.4171/jems/1269 COLLEGE NANME COLLEGE CODE Swansea University Supported in part by the National Key R&D Program of China (No. 2020YFA0712900) and NNSFC (11831014, 11921001). 2025-05-15T16:04:15.8056318 2022-07-19T11:54:39.1002150 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics Feng-yu Wang 1 60533__24634__78aa3871522549abbb6e596707fbc782.pdf 60533.pdf 2022-07-19T11:57:46.1178606 Output 373635 application/pdf Accepted Manuscript true true eng
title Convergence in Wasserstein distance for empirical measures of Dirichlet diffusion processes on manifolds
spellingShingle Convergence in Wasserstein distance for empirical measures of Dirichlet diffusion processes on manifolds
Feng-yu Wang
title_short Convergence in Wasserstein distance for empirical measures of Dirichlet diffusion processes on manifolds
title_full Convergence in Wasserstein distance for empirical measures of Dirichlet diffusion processes on manifolds
title_fullStr Convergence in Wasserstein distance for empirical measures of Dirichlet diffusion processes on manifolds
title_full_unstemmed Convergence in Wasserstein distance for empirical measures of Dirichlet diffusion processes on manifolds
title_sort Convergence in Wasserstein distance for empirical measures of Dirichlet diffusion processes on manifolds
author_id_str_mv 6734caa6d9a388bd3bd8eb0a1131d0de
author_id_fullname_str_mv 6734caa6d9a388bd3bd8eb0a1131d0de_***_Feng-yu Wang
author Feng-yu Wang
author2 Feng-yu Wang
format Journal article
container_title Journal of the European Mathematical Society
container_volume 25
container_issue 9
publishDate 2022
institution Swansea University
issn 1435-9855
1435-9863
doi_str_mv 10.4171/jems/1269
publisher European Mathematical Society - EMS - Publishing House GmbH
college_str Faculty of Science and Engineering
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hierarchy_top_id facultyofscienceandengineering
hierarchy_top_title Faculty of Science and Engineering
hierarchy_parent_id facultyofscienceandengineering
hierarchy_parent_title Faculty of Science and Engineering
department_str School of Mathematics and Computer Science - Mathematics{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Mathematics and Computer Science - Mathematics
document_store_str 1
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description Let M be a d-dimensional connected compact Riemannian manifold with boundary ∂M, let V∈C2(M) such that μ(dx):=eV(x)dx is a probability measure, and let Xt be the diffusion process generated by L:=Δ+∇V with τ:=inf{t≥0:Xt∈∂M}. Consider the empirical measure μt:=1t∫t0δXsds under the condition t<τ for the diffusion process. If d≤3, then for any initial distribution not fully supported on ∂M,c∑m=1∞2(λm−λ0)2≤lim inft→∞infT≥t{tE[W2(μt,μ0)2∣∣T<τ]}≤lim supt→∞supT≥t{tE[W2(μt,μ0)2∣∣T<τ]}≤∑m=1∞2(λm−λ0)2holds for some constant c∈(0,1] with c=1 when ∂M is convex, where μ0:=ϕ20μ for the first Dirichet eigenfunction ϕ0 of L, {λm}m≥0 are the Dirichlet eigenvalues of −L listed in the increasing order counting multiplicities, and the upper bound is finite if and only if d≤3. When d=4, supT≥tE[W2(μt,μ0)2∣∣T<τ] decays in the order t−1logt, while for d≥5 it behaves like t−2d−2, as t→∞.
published_date 2022-09-03T05:05:01Z
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score 11.089572

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