Complete Logarithmic Sobolev inequalities via Ricci curvature bounded below

TL;DR

This paper establishes conditions under which Ricci curvature bounds and mixing time estimates imply modified log-Sobolev inequalities for Markov semigroups, including quantum and matrix-valued cases.

Contribution

It proves that Ricci curvature bounds combined with mixing time estimates imply modified log-Sobolev inequalities for a broad class of Markov semigroups, including quantum and matrix-valued extensions.

Findings

Ricci curvature bounded below plus finite L_infinity-mixing time implies modified log-Sobolev inequality.

Results apply to non-ergodic quantum Markov semigroups with noncommutative Ricci bounds.

Heat semigroup on compact manifolds admits uniform modified log-Sobolev inequalities for matrix extensions.

Abstract

We prove that for a symmetric Markov semigroup, Ricci curvature bounded from below by a non-positive constant combined with a finite $L_\infty$-mixing time implies the modified log-Sobolev inequality. Such $L_\infty$-mixing time estimates always hold for Markov semigroups that have spectral gap and finite Varopoulos dimension. Our results apply to non-ergodic quantum Markov semigroups with noncommutative Ricci curvature bounds recently introduced by Carlen and Maas. As an application, we prove that the heat semigroup on a compact Riemannian manifold admits a uniform modified log-Sobolev inequality for all its matrix-valued extensions.