On the variational interpretation of local logarithmic Sobolev inequalities

Gauthier Clerc (1) , Giovanni Conforti (2) , Ivan Gentil (1)

1 ICJ - Institut Camille Jordan
2 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique

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 [JKO98]. However, the local versions of such inequalities, which can be proven by means of Bakry-Émery-Ledoux Γ calculus, has not yet been given an interpretation in terms of this Riemannian formalism. In this short note we close this gap by explaining heuristically 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.

Domaines: Analyse classique [math.CA]; Probabilités [math.PR]

Gauthier Clerc, Giovanni Conforti, Ivan Gentil. On the variational interpretation of local logarithmic Sobolev inequalities. 2021. ⟨hal-02996841v2⟩