Bakry-Émery calculus for entropic curvature, new diameter estimates, and spectral gaps

2023/12/15 by Kamtue, Supanat, Liu, Shiping, Münch, Florentin +1
#05C10 #51K10 (Secondary) #53C21 (Primary) 53A70 #Combinatorics (math.CO) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2312.09686

Abstract

In this paper, we propose a generalization of Bakry-Émery's calculus which allows us to formulate both Bakry-Émery and entropic curvature simultaneously. This formulation represents both curvatures as an integral of the Bochner formula against some measure. This leads to a natural optimality criterion of measures, which we investigate in the Bakry-Émery setting. Moreover, our approach leads also to a dimension parameter in the framework of entropic curvature. We also present gradient estimate applications, that is, diameter estimates for Markov chains with strictly positive entropic curvature and a spectral gap estimate. The latter implies non-existence of non-negatively curved expanders.

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