# On the variational interpretation of local logarithmic Sobolev inequalities

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 Schrödinger problem yields a variations interpretation of the local logarithmic Sobolev inequalities, that could possibly unlock novel class of local inequalities.
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https://hal.archives-ouvertes.fr/hal-02996841
Contributor : Ivan Gentil Connect in order to contact the contributor
Submitted on : Sunday, October 24, 2021 - 4:51:07 PM
Last modification on : Tuesday, October 26, 2021 - 4:00:39 AM

### Files

HAL-Bakry vs Otto-4.pdf
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### Identifiers

• HAL Id : hal-02996841, version 3
• ARXIV : 2011.05207

### Citation

Gauthier Clerc, Giovanni Conforti, Ivan Gentil. On the variational interpretation of local logarithmic Sobolev inequalities. 2021. ⟨hal-02996841v3⟩

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