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Transportation onto log-Lipschitz perturbations

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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.

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Notes

  1. The existence of solutions to the above equations, and of the limits, is non-trivial in general. These issues are discussed in length in [1, 14, 24] for the various settings we consider. In particular, the convexity and curvature assumptions we shall enforce in the coming proofs are enough to guarantee that all maps under consideration are well-defined.

  2. We rescale the generator of the heat equation so our t corresponds to 2t in [28].

  3. We rescale the generator of the heat equation so our t corresponds to 2t in [28].

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Acknowledgements

We gratefully acknowledge the contributions of Joe Neeman who was involved in the earlier stages of this work. In particular, the example showing Theorem 1 is sharp is due to him. We also thank Patrick Cattiaux, Daniel Lacker, Michel Ledoux, and Lorenzo Schiavo for useful discussions. In addition, we thank the ananoymous referee for their helpful comments. M.F. was supported by the Projects MESA (ANR-18-CE40-006) and EFI (ANR-17-CE40-0030) of the French National Research Agency (ANR). This material is based upon work supported by the National Science Foundation under Award Number 2002022.

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Authors and Affiliations

  1. Laboratoire Jacques Louis Lions & Laboratoire de Probabilités Statistique et Modélisation, Université Paris Cité and Sorbonne Université and CNRS, 75013, Paris, France

    Max Fathi

  2. Département de Mathématiques et applications, Ecole Normale Supérieure, Université PSL, 75005, Paris, France

    Max Fathi

  3. Institut Universitaire de France (IUF), Paris, France

    Max Fathi

  4. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA

    Dan Mikulincer

  5. Division of Applied Mathematics, Brown University, Providence, RI, USA

    Yair Shenfeld

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  1. Max Fathi
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Correspondence to Max Fathi.

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Communicated by Andrea Mondino.

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Fathi, M., Mikulincer, D. & Shenfeld, Y. Transportation onto log-Lipschitz perturbations. Calc. Var. 63, 61 (2024). https://doi.org/10.1007/s00526-023-02652-x

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