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Log-Hessian and Deviation Bounds for Markov Semi-Groups, and Regularization Effect in \(\mathbb {L}^{1}\)

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Abstract

It is well known that some important Markov semi-groups have a “regularization effect” – as for example th hypercontractivity property of the noise operator on the Boolean hypercube or the Ornstein-Uhlenbeck semi-group on the real line, which applies to functions in Lp for p > 1. Talagrand had conjectured in 1989 that the noise operator on the Boolean hypercube has a further subtle regularization property for functions that are just integrable, but this conjecture remains open. Nonetheless, the Gaussian analogue of this conjecture was proven in recent years by Eldan-Lee and Lehec, by combining an inequality for the log-Hessian of the Ornstein-Uhlenbeck semi-group with a new deviation inequality for log-semi-convex functions under Gaussian measure. In this work, we explore the question of how much more general this phenomenon is. Specifically, our first goal is to explore the validity of both these ingredients for some diffusion semi-groups in \(\mathbb {R}^{n}\), as well as for the \(M/M/\infty \) queue on the non-negative integers and the Laguerre semi-groups on the positive real line. Our second goal is to prove a one-dimensional regularization effect for these settings, even in those cases where these ingredients are not valid.

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Acknowledgements

We thank Zhen-Qing Chen for enlightening discussions on the topic of this paper and an anonymous referee for his/her constructive comments.

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Authors and Affiliations

  1. CNRS, Université de Paris, MAP5 UMR 8145, F-75006, Paris, France

    N. Gozlan

  2. Department of Mathematics, Imperial College London, London, SW7 2AZ, UK

    Xue-Mei Li

  3. Department of Mathematical Sciences, University of Delaware, 501 Ewing Hall, Newark, DE, 19716, USA

    M. Madiman

  4. Modal’X, FP2M, Université Paris Nanterre, CNRS FR 2036, 200 avenue de la République, 92000, Nanterre, France

    C. Roberto

  5. LAMA, University of Gustave Eiffel, UPEM, Univ Paris Est Creteil, CNRS, F-77447, Marne-la-Vallée, France

    P.-M. Samson

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  1. N. Gozlan
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  4. C. Roberto
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Correspondence to C. Roberto.

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Supported by the grants ANR-15-CE40-0020-03 -LSD -Large Stochastic Dynamics; ANR 10-LABX-0058 -Labex Bezout; ANR 11-LBX-0023-01 -Labex MME-DII and the grant DMS-1409504 from the U.S. National Science Foundation. Nathael Gozlan, Cyril Roberto and Paul-Marie Samson are supported by a grant of the Simone and Cino Del Duca foundation. This research has been conducted within the FP2M federation (CNRS FR 2036). Li’s research is partially supported by EP/S023925/1.

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Gozlan, N., Li, XM., Madiman, M. et al. Log-Hessian and Deviation Bounds for Markov Semi-Groups, and Regularization Effect in \(\mathbb {L}^{1}\). Potential Anal 58, 123–158 (2023). https://doi.org/10.1007/s11118-021-09934-z

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  • DOI: https://doi.org/10.1007/s11118-021-09934-z

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