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Exponential Contraction in Wasserstein Distances for Diffusion Semigroups with Negative Curvature

Feng-yu Wang

Potential Analysis, Volume: 53, Issue: 3, Pages: 1123 - 1144

Swansea University Author: Feng-yu Wang

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DOI (Published version): 10.1007/s11118-019-09800-z

Abstract

Let Pt be the (Neumann) diffusion semigroup Pt generated by a weighted Laplacian on acomplete connected Riemannian manifold M without boundary or with a convex boundary.It is well known that the Bakry-Emery curvature is bounded below by a positive constantλ > 0 if and only ifWp(μ1Pt, μ2Pt) ≤ e−λt...

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Published in: Potential Analysis
ISSN: 0926-2601 1572-929X
Published: Springer Science and Business Media LLC 2020
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URI: https://cronfa.swan.ac.uk/Record/cronfa51763
first_indexed 2019-09-10T15:31:36Z
last_indexed 2023-03-14T04:05:16Z
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spelling 2023-03-13T12:01:24.1051103 v2 51763 2019-09-10 Exponential Contraction in Wasserstein Distances for Diffusion Semigroups with Negative Curvature 6734caa6d9a388bd3bd8eb0a1131d0de Feng-yu Wang Feng-yu Wang true false 2019-09-10 Let Pt be the (Neumann) diffusion semigroup Pt generated by a weighted Laplacian on acomplete connected Riemannian manifold M without boundary or with a convex boundary.It is well known that the Bakry-Emery curvature is bounded below by a positive constantλ > 0 if and only ifWp(μ1Pt, μ2Pt) ≤ e−λtWp(μ1, μ2), t ≥ 0, p ≥ 1holds for all probability measures μ1 and μ2 on M, where Wp is the Lp Wasserstein distanceinduced by the Riemannian distance. In this paper, we prove the exponential contractionWp(μ1Pt, μ2Pt) ≤ ce−λtWp(μ1, μ2), p ≥ 1, t ≥ 0for some constants c, λ > 0 for a class of diffusion semigroups with negative curvaturewhere the constant c is essentially larger than 1. Similar results are derived for SDEs withmultiplicative noise by using explicit conditions on the coefficients, which are new even forSDEs with additive noise. Journal Article Potential Analysis 53 3 1123 1144 Springer Science and Business Media LLC 0926-2601 1572-929X Wasserstein distance; Diffusion semigroup; Riemannian manifold; Curvature condition; SDEs with multiplicative noise 1 10 2020 2020-10-01 10.1007/s11118-019-09800-z http://dx.doi.org/10.1007/s11118-019-09800-z COLLEGE NANME COLLEGE CODE Swansea University 2023-03-13T12:01:24.1051103 2019-09-10T12:37:46.4282785 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics Feng-yu Wang 1 51763__15213__b5ae057210e940d4a1766a61e8987931.pdf 16a.pdf 2019-09-10T12:40:59.3930000 Output 350600 application/pdf Accepted Manuscript true 2021-02-06T00:00:00.0000000 true eng
title Exponential Contraction in Wasserstein Distances for Diffusion Semigroups with Negative Curvature
spellingShingle Exponential Contraction in Wasserstein Distances for Diffusion Semigroups with Negative Curvature
Feng-yu Wang
title_short Exponential Contraction in Wasserstein Distances for Diffusion Semigroups with Negative Curvature
title_full Exponential Contraction in Wasserstein Distances for Diffusion Semigroups with Negative Curvature
title_fullStr Exponential Contraction in Wasserstein Distances for Diffusion Semigroups with Negative Curvature
title_full_unstemmed Exponential Contraction in Wasserstein Distances for Diffusion Semigroups with Negative Curvature
title_sort Exponential Contraction in Wasserstein Distances for Diffusion Semigroups with Negative Curvature
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 Potential Analysis
container_volume 53
container_issue 3
container_start_page 1123
publishDate 2020
institution Swansea University
issn 0926-2601
1572-929X
doi_str_mv 10.1007/s11118-019-09800-z
publisher Springer Science and Business Media LLC
college_str Faculty of Science and Engineering
hierarchytype
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
url http://dx.doi.org/10.1007/s11118-019-09800-z
document_store_str 1
active_str 0
description Let Pt be the (Neumann) diffusion semigroup Pt generated by a weighted Laplacian on acomplete connected Riemannian manifold M without boundary or with a convex boundary.It is well known that the Bakry-Emery curvature is bounded below by a positive constantλ > 0 if and only ifWp(μ1Pt, μ2Pt) ≤ e−λtWp(μ1, μ2), t ≥ 0, p ≥ 1holds for all probability measures μ1 and μ2 on M, where Wp is the Lp Wasserstein distanceinduced by the Riemannian distance. In this paper, we prove the exponential contractionWp(μ1Pt, μ2Pt) ≤ ce−λtWp(μ1, μ2), p ≥ 1, t ≥ 0for some constants c, λ > 0 for a class of diffusion semigroups with negative curvaturewhere the constant c is essentially larger than 1. Similar results are derived for SDEs withmultiplicative noise by using explicit conditions on the coefficients, which are new even forSDEs with additive noise.
published_date 2020-10-01T07:37:46Z
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score 11.057796

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