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Stability of Talagrand’s Gaussian transport-entropy inequality via the Föllmer process

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Abstract

We establish a dimension-free improvement of Talagrand’s Gaussian transport-entropy inequality, under the assumption that the measures satisfy a Poincaré inequality. We also study stability of the inequality, in terms of relative entropy, when restricted to measures whose covariance matrix trace is smaller than the ambient dimension. In case the covariance matrix is strictly smaller than the identity, we give dimension-free estimates which depend on its eigenvalues. To complement our results, we show that our conditions cannot be relaxed, and that there exist measures with covariance larger than the identity, for which the inequality is not stable, in relative entropy. To deal with these examples, we show that, without any assumptions, one can always get quantitative stability estimates in terms of relative entropy to Gaussian mixtures. Our approach gives rise to a new point of view which sheds light on the hierarchy between Fisher information, entropy and transportation distance, and may be of independent interest. In particular, it implies that the described results apply verbatim to the log-Sobolev inequality and improve upon some known estimates in the literature.

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Acknowledgments

We wish to thank Ronen Eldan, Max Fathi, Renan Gross, Emanuel Indrei and Yair Shenfeld for useful discussions and for their comments concerning a preliminary draft of this work. We are also grateful to the anonymous referee for carefully reading this paper and providing thoughtful comments.

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Authors and Affiliations

  1. Weizmann Institute of Science, Herzl 234, Rehovot, 7610001, Israel

    Dan Mikulincer

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  1. Dan Mikulincer
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Correspondence to Dan Mikulincer.

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Supported by an Azrieli foundation fellowship.

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Mikulincer, D. Stability of Talagrand’s Gaussian transport-entropy inequality via the Föllmer process. Isr. J. Math. 242, 215–241 (2021). https://doi.org/10.1007/s11856-021-2129-x

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  • DOI: https://doi.org/10.1007/s11856-021-2129-x

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