Archive for Rational Mechanics and Analysis

, Volume 206, Issue 3, pp 997–1038 | Cite as

Ricci Curvature of Finite Markov Chains via Convexity of the Entropy

  • Matthias Erbar
  • Jan Maas


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–Émery and Otto–Villani. Further, 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.


Entropy Markov Chain Ricci Curvature Discrete Analogue Simple Random Walk 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Institute for Applied MathematicsUniversity of BonnBonnGermany

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