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arXiv · 2011.05207

On the variational interpretation of local logarithmic Sobolev inequalities

Gauthier Clerc·Giovanni Conforti·Ivan Gentil
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Abstract

The celebrated Otto calculus has established itself as a powerful tool for proving quantitative energy dissipation estimates and provides with an elegant geometric interpretation of certain functional inequalities such as the Logarithmic Sobolev inequality. However, the \emph{local} versions of such inequalities, which can be proven by means of Bakry-Emery-Ledoux $\Gamma$-calculus, has not yet been given an interpretation in terms of this Riemannian formalism. In this short note we close this gap by explaining how Otto calculus applied to the Schr{\"o}dinger problem yields a variations interpretation of the local logarithmic Sobolev inequalities, that could possibly unlock novel class of local inequalities.

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Gauthier Clerc, Giovanni Conforti, Ivan Gentil. 2020-11-10. On the variational interpretation of local logarithmic Sobolev inequalities. https://arxiv.org/abs/2011.05207

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