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Exponential Contraction in Wasserstein Distances for Diffusion Semigroups with Negative Curvature

Feng-yu Wang Orcid Logo

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

Swansea University Author: Feng-yu Wang Orcid Logo

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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
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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−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.
Keywords: Wasserstein distance; Diffusion semigroup; Riemannian manifold; Curvature condition; SDEs with multiplicative noise
College: Faculty of Science and Engineering
Issue: 3
Start Page: 1123
End Page: 1144

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