Ricci curvature of finite Markov chains via convexity of the entropy

M. Erbar, J. Maas, Archive for Rational Mechanics and Analysis 206 (2012) 997–1038.


Journal Article | Published
Author
Erbar, Matthias; Maas, JanIST Austria
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–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.
Publishing Year
Date Published
2012-12-01
Journal Title
Archive for Rational Mechanics and Analysis
Acknowledgement
Jan Maas is supported by Rubicon subsidy 680-50-0901 of the Netherlands Organisation for Scientific Research (NWO).
Volume
206
Issue
3
Page
997 - 1038
IST-REx-ID

Cite this

Erbar M, Maas J. Ricci curvature of finite Markov chains via convexity of the entropy. Archive for Rational Mechanics and Analysis. 2012;206(3):997-1038. doi:10.1007/s00205-012-0554-z
Erbar, M., & Maas, J. (2012). Ricci curvature of finite Markov chains via convexity of the entropy. Archive for Rational Mechanics and Analysis, 206(3), 997–1038. https://doi.org/10.1007/s00205-012-0554-z
Erbar, Matthias, and Jan Maas. “Ricci Curvature of Finite Markov Chains via Convexity of the Entropy.” Archive for Rational Mechanics and Analysis 206, no. 3 (2012): 997–1038. https://doi.org/10.1007/s00205-012-0554-z.
M. Erbar and J. Maas, “Ricci curvature of finite Markov chains via convexity of the entropy,” Archive for Rational Mechanics and Analysis, vol. 206, no. 3, pp. 997–1038, 2012.
Erbar M, Maas J. 2012. Ricci curvature of finite Markov chains via convexity of the entropy. Archive for Rational Mechanics and Analysis. 206(3), 997–1038.
Erbar, Matthias, and Jan Maas. “Ricci Curvature of Finite Markov Chains via Convexity of the Entropy.” Archive for Rational Mechanics and Analysis, vol. 206, no. 3, Springer, 2012, pp. 997–1038, doi:10.1007/s00205-012-0554-z.
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