{"date_published":"2012-12-01T00:00:00Z","abstract":[{"text":"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.","lang":"eng"}],"page":"997 - 1038","date_created":"2018-12-11T11:55:52Z","date_updated":"2021-01-12T06:55:28Z","doi":"10.1007/s00205-012-0554-z","title":"Ricci curvature of finite Markov chains via convexity of the entropy","quality_controlled":0,"acknowledgement":"Jan Maas is supported by Rubicon subsidy 680-50-0901 of the Netherlands Organisation for Scientific Research (NWO).","author":[{"first_name":"Matthias","full_name":"Erbar, Matthias","last_name":"Erbar"},{"id":"4C5696CE-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-0845-1338","last_name":"Maas","first_name":"Jan","full_name":"Jan Maas"}],"month":"12","intvolume":"       206","publication_status":"published","oa":1,"main_file_link":[{"url":"http://arxiv.org/abs/1111.2687","open_access":"1"}],"publist_id":"4906","publisher":"Springer","issue":"3","type":"journal_article","citation":{"ieee":"M. Erbar and J. Maas, “Ricci curvature of finite Markov chains via convexity of the entropy,” <i>Archive for Rational Mechanics and Analysis</i>, vol. 206, no. 3. Springer, pp. 997–1038, 2012.","ista":"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.","chicago":"Erbar, Matthias, and Jan Maas. “Ricci Curvature of Finite Markov Chains via Convexity of the Entropy.” <i>Archive for Rational Mechanics and Analysis</i>. Springer, 2012. <a href=\"https://doi.org/10.1007/s00205-012-0554-z\">https://doi.org/10.1007/s00205-012-0554-z</a>.","apa":"Erbar, M., &#38; Maas, J. (2012). Ricci curvature of finite Markov chains via convexity of the entropy. <i>Archive for Rational Mechanics and Analysis</i>. Springer. <a href=\"https://doi.org/10.1007/s00205-012-0554-z\">https://doi.org/10.1007/s00205-012-0554-z</a>","ama":"Erbar M, Maas J. Ricci curvature of finite Markov chains via convexity of the entropy. <i>Archive for Rational Mechanics and Analysis</i>. 2012;206(3):997-1038. doi:<a href=\"https://doi.org/10.1007/s00205-012-0554-z\">10.1007/s00205-012-0554-z</a>","short":"M. Erbar, J. Maas, Archive for Rational Mechanics and Analysis 206 (2012) 997–1038.","mla":"Erbar, Matthias, and Jan Maas. “Ricci Curvature of Finite Markov Chains via Convexity of the Entropy.” <i>Archive for Rational Mechanics and Analysis</i>, vol. 206, no. 3, Springer, 2012, pp. 997–1038, doi:<a href=\"https://doi.org/10.1007/s00205-012-0554-z\">10.1007/s00205-012-0554-z</a>."},"day":"01","volume":206,"extern":1,"year":"2012","status":"public","publication":"Archive for Rational Mechanics and Analysis","_id":"2127"}