Transportation onto log-Lipschitz perturbations
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Author(s)
Fathi, Max
Mikulincer, Dan
Shenfeld, Yair
Date Issued
February 20, 2024
Journal
Calculus of Variations and Partial Differential Equations
Publisher
Springer Berlin Heidelberg
Citation
Fathi, M., Mikulincer, D. & Shenfeld, Y. Transportation onto log-Lipschitz perturbations. Calc. Var. 63, 61 (2024).
Version
Author's final manuscript
Abstract
We establish sufficient conditions for the existence of globally Lipschitz transport maps between probability measures and their log-Lipschitz perturbations, with dimension-free bounds. Our results include Gaussian measures on Euclidean spaces and uniform measures on spheres as source measures. More generally, we prove results for source measures on manifolds satisfying strong curvature assumptions. These seem to be the first examples of dimension-free Lipschitz transport maps in non-Euclidean settings, which are moreover sharp on the sphere. We also present some applications to functional inequalities, including a new dimension-free Gaussian isoperimetric inequality for log-Lipschitz perturbations of the standard Gaussian measure. Our proofs are based on the Langevin flow construction of transport maps of Kim and Milman.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
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DOI of Published Version
https://doi.org/10.1007/s00526-023-02652-x
