Ricci Curvature of Finite Markov Chains via Convexity of the Entropy

By Matthias Erbar, Jan Maas. 2011

243 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/c7a79d7843277b7f5614b62cbdbb8d706e12111e

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Inference for empirical Wasserstein distances on finite spaces (2016, background): “The Wasserstein distance on finite spaces has been considered recently by Gozlan et al. (2013) to derive entropy inequalities on graphs and by Erbar and Maas (2012) to define Ricci curvature for Markov chains on discrete spaces.”
On quotients of spaces with Ricci curvature bounded below (2017, methodology): “Definition A.6 (Maas-Mielke Ricci curvature).”
On the difference between entropic cost and the optimal transport cost (2019, unknown): “See articles [21, 10, 8, 11, 6, 9].”
Curvature and transport inequalities for Markov chains in discrete spaces (2015, background): “Erbar and Maas recovered some of those functional inequality results with the notion of entropic Ricci curvature on graphs, and we refer the reader to [14,32] for more details.”
Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance (2016, background): “Related work in the commutative setting can be found in [22, 23, 27, 44, 46, 48].”
Discrete Curvature and Abelian Groups (2015, background): “approaches to discrete Ricci curvature and related inequalities, see [5, 12, 15, 20, 22, 24, 25].”
Gradient flows and Evolution Variational Inequalities in metric spaces. I: Structural properties (2018, unknown): “(P2(X), dW ), where X is an RCD(K,∞) metric-measure space and φ is the relative entropy functional [88, 69], and there is an intensive research to construct EVI gradient flows in ad-hoc geometric settings for reaction-diffusion equations [54, 25, 57, 55] and systems [65, 48, 56], nonlinear viscoelas”
Kantorovich duality for general transport costs and applications (2014, result): “We refer to the works by Ollivier [47], Bonciocat-Sturm [14], Ollivier-Villani [48], Erbar-Maas [21], Hillion [34] and the work [30] by the authors for different attempts to give a meaning to the notion of “discrete curvature”.”
Displacement convexity of entropy and related inequalities on graphs (2012, background): “In [12], Erbar and Maas introduce a pseudoWasserstein distance W2 that corresponds to the geodesic distance on the set, P(Ωn), of probability measures on the hypercube Ωn, equipped with a Riemannian metric.”
Gradient flows of the entropy for jump processes (2012, methodology): “This could be exploited to derive new functional inequalities and rates of convergence to equilibrium for the corresponding evolution equation, as has been done in the discrete setting of finite Markov chains in [16].”