Geodesic convexity of the relative entropy in reversible Markov chains

By Alexander Mielke. 2013

186 citations recorded by the source corpus.

Subfields: Applied Mathematics, Astronomy and Astrophysics

View Paper: https://www.semanticscholar.org/paper/5730729a3f2d6a5928bfa734e5dc4393a8b5acbe

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In Applied Mathematics

In Astronomy and Astrophysics

Citations in the corpus

Displacement convexity of entropy and related inequalities on graphs (2012, background): “However, while a lot is now known in the Riemannian setting (and more generally in geodesic spaces), very little is known so f ar in discrete spaces (such as finite graphs or finite Markov chains), with the notable exception of some n otions of (discrete) Ricci curvature proposed recently by several”
Gradient flows of the entropy for jump processes (2012, background): “Very recently, Maas [20] (see also [22], [10] for independent related work by Mielke and Chow et al.) introduced a distance between probability measures on a discrete space equipped with a Markov kernel such that the law of the continuous time Markov chain evolves as the gradient ﬂow of the entropy.”
An Analog of the 2-Wasserstein Metric in Non-Commutative Probability Under Which the Fermionic Fokker–Planck Equation is Gradient Flow for the Entropy (2012, background): “A similar gradient flow structure in the space of probability measures has meanwhile been developed and exploited in many different settings [1, 2, 6, 9, 10, 12, 14, 15, 20, 22, 23, 25, 26].”
Jump processes as generalized gradient flows (2020, background): “[LMS18] M. Liero, A. Mielke, and G. Savaré.”
On quotients of spaces with Ricci curvature bounded below (2017, methodology): “Definition A.6 (Maas-Mielke Ricci curvature).”
Ricci Curvature of Finite Markov Chains via Convexity of the Entropy (2011, background): “After essentially finishing this paper, the authors have been informed about the preprint [30], in which geodesic convexity of the entropy for Markov chains has been studied as well.”
Curvature and transport inequalities for Markov chains in discrete spaces (2015, background): “Other notions that have been developed (and which we shall not discuss further here) include the entropic Ricci curvature defined in [14] and [34], which is based on the Lott–Sturm–Villani definition of curvature [30,43], geodesic convexity along interpolations in [17] and [28], rough curvature boun”
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].”
Gradient flow structures for discrete porous medium equations (2012, background): “tropy functionals. Of crucial importance in the theory of Wasserstein gradient ﬂows is that the relevant entropy functionals exhibit good convexity properties along Wasserstein geodesics. Recent work [8, 13] shows that analogous properties hold for the relative entropy functional H in some discrete ”