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 [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.

Gradient estimates for the Schrödinger potentials: convergence to the Brenier map and quantitative stability

“…The backbone of our proof strategy are gradient estimates for Schr€ odinger potentials, that we call corrector estimates in view of the above stochastic control interpretation, since they provide contractive estimates for ka P ðt, xÞk L 2 ðdPÞ for t 2 ½0, T: These bounds have been put forward in [11] to show a quantitative form of convex entropy decay along entropic interpolations and are here proven under much weaker assumptions. So far, corrector estimates have found applications in the proof of new functional inequalities and in the study of the long-time behavior of entropic interpolations, see [11,12] for example, and are known to be equivalent to the celebrated Bakry-Emery condition, see [13]. However, to the best of our knowledge, the results of this article are the first to show the interest of such bounds in the analysis of the convergence of SP toward OT, in which one has to deal with short-time limits instead of large-time limits.…”