2010/05/08 by Shin-ichi Ohta, Shin‐ichi Ohta, Ohta, Shin-ichi +2
Mathematics · Physics and Astronomy · #49 #53 #58 #Advanced Differential Geometry Research #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Probability (math.PR) #math.AP #math.DG #math.PR #msc:49 #msc:53 #msc:58
paper · pdf · doi:10.48550/arxiv.1005.1331
43pages
openalex publication_date 2010/05/08 · arxiv created 2011/07/04 · arxiv updated 2011/07/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.