2024/04/06 by Supanat Kamtue, Kamtue, Supanat, Shiping Liu +5
Engineering · Mathematics · Physics and Astronomy · #05C81 #53C21 #60J10 #Advanced Differential Geometry Research #Differential Geometry (math.DG) #Elasticity and Material Modeling #FOS: Mathematics #Geometric Analysis and Curvature Flows #Metric Geometry (math.MG) #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.2404.04581
openalex publication_date 2024/04/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we consider global θ-curvatures of finite Markov chains with associated means θ in the spirit of the entropic curvature (based on the logarithmic mean) by Erbar-Maas and Mielke. As in the case of Bakry-Émery curvature, we also allow for a finite dimension parameter by making use of an adapted Γ calculus for θ-curvatures. We prove explicit positive lower curvature bounds (both finite- and infinite-dimensional) for finite abelian Cayley graphs. In the case of cycles, we provide also an upper curvature bound which shows that our lower bounds are asymptotically sharp (up to a logarithmic factor). Moreover, we prove new universal lower curvature bounds for finite Markov chains as well as curvature perturbation results (allowing, in particular, to compare entropic and Bakry-Émery curvatures). Finally, we present examples where entropic curvature differs significantly from other curvature notions like Bakry-Émery curvature or Ollivier Ricci and sectional curvatures.