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arXiv · 2007.06138

Complete Logarithmic Sobolev inequalities via Ricci curvature bounded below

Michael Brannan·Li Gao·Marius Junge
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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.

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Michael Brannan, Li Gao, Marius Junge. 2020-07-13. Complete Logarithmic Sobolev inequalities via Ricci curvature bounded below. https://arxiv.org/abs/2007.06138

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