Ricci curvature of finite Markov chains via convexity of the entropy

Abstract

We study a new notion of Ricci curvature that applies to Markov chains on discrete spaces. This notion relies on geodesic convexity of the entropy and is analogous to the one introduced by Lott, Sturm, and Villani for geodesic measure spaces. In order to apply to the discrete setting, the role of the Wasserstein metric is taken over by a different metric, having the property that continuous time Markov chains are gradient flows of the entropy. Using this notion of Ricci curvature we prove discrete analogues of fundamental results by Bakry--Emery and Otto--Villani. Furthermore we show that Ricci curvature bounds are preserved under tensorisation. As a special case we obtain the sharp Ricci curvature lower bound for the discrete hypercube.

39 pages, to appear in Arch. Ration. Mech. Anal

Cited by in corpus (29)

• Non-commutative calculus, optimal transport and functional inequalities in dissipative quantum systems
• Entropic Ricci curvature bounds for discrete interacting systems
• Housekeeping and excess entropy production for general nonlinear dynamics
• An entropic gradient structure for Lindblad equations and couplings of quantum systems to macroscopic models
• Gradient flow structure for McKean-Vlasov equations on discrete spaces
• On quotients of spaces with Ricci curvature bounded below
• The Graph Curvature Calculator and the curvatures of cubic graphs
• A discrete log-Sobolev inequality under a Bakry-Emery type condition
• Emergence of the Circle in a Statistical Model of Random Cubic Graphs
• The Heat Flow on Metric Random Walk Spaces
• Population games and Discrete optimal transport
• On microscopic origins of generalized gradient structures
• Long-time behavior of a finite volume discretization for a fourth order diffusion equation
• Distance bounds for graphs with some negative Bakry-Émery curvature
• Discrete versions of the transport equation and the Shepp-Olkin conjecture
• Entropy decay for interacting systems via the Bochner--Bakry--Emery approach
• Complete gradient estimates of quantum Markov semigroups
• Augmentations of Forman's Ricci Curvature and their Applications in Community Detection
• Entropic curvature and convergence to equilibrium for mean-field dynamics on discrete spaces
• The Equilibrium States of Large Networks of Erlang Queues
• Perpetual cutoff method and discrete Ricci curvature bounds with exceptions
• Entropy and thinning of discrete random variables
• Absolutely continuous and BV-curves in 1-Wasserstein spaces
• Wasserstein geometry and Ricci curvature bounds for Poisson spaces
• Discrete-to-continuum limits of optimal transport with linear growth on periodic graphs
• Coarse extrinsic curvature of Riemannian submanifolds
• Contractive coupling rates and curvature lower bounds for Markov chains
• Exploring the space of graphs with fixed discrete curvatures
• Stochastic homogenization of dynamical discrete optimal transport