Complete logarithmic Sobolev inequalities via Ricci curvature bounded below

Michael Brannan, Li Gao, Marius Junge

Research output: Contribution to journalArticlepeer-review


We prove that for a symmetric Markov semigroup, Ricci curvature bounded from below by a non-positive constant combined with a finite L-mixing time implies the modified log-Sobolev inequality. Such L-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.

Original languageEnglish (US)
Article number108129
JournalAdvances in Mathematics
StatePublished - Jan 22 2022


  • Heat semigroup
  • Logarithmic Sobolev inequality
  • Quantum Markov semigroup
  • Ricci curvature

ASJC Scopus subject areas

  • General Mathematics


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