Displacement convexity of generalized relative entropies

Shin ichi Ohta, Asuka Takatsu

Research output: Contribution to journalArticlepeer-review

17 Citations (Scopus)

Abstract

We investigate the m-relative entropy, which stems from the Bregman divergence, on weighted Riemannian and Finsler manifolds. We prove that the displacement K-convexity of the m-relative entropy is equivalent to the combination of the nonnegativity of the weighted Ricci curvature and the K-convexity of the weight function. We use this to show appropriate variants of the Talagrand, HWI and the logarithmic Sobolev inequalities, as well as the concentration of measures. We also prove that the gradient flow of the m-relative entropy produces a solution to the porous medium equation or the fast diffusion equation.

Original languageEnglish
Pages (from-to)1742-1787
Number of pages46
JournalAdvances in Mathematics
Volume228
Issue number3
DOIs
Publication statusPublished - 2011 Oct 20
Externally publishedYes

Keywords

  • Bregman divergence
  • Porous medium equation
  • Relative entropy
  • Ricci curvature
  • Wasserstein space

ASJC Scopus subject areas

  • Mathematics(all)

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