Mass transportation and rough curvature bounds for discrete spaces

By Anca I. Bonciocat, Karl‐Theodor Sturm. 2009

98 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computational Mechanics

View Paper: https://www.semanticscholar.org/paper/a39ba5e787d64702181be49f9f67aa76e31cb448

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In Applied Mathematics

In Computational Mechanics

Citations in the corpus

Ollivier-Ricci curvature and the spectrum of the normalized graph Laplace operator (2011, background): “By now, there exist many notions of generalized curvature, and several of them have found important applications, see Sturm [24], Lott–Villani [19], Ollivier [21], Ohta [20], Bonciocat–Sturm [3], Joulin–Ollivier [14] and the references therein.”
Localization and Tensorization Properties of the Curvature-Dimension Condition for Metric Measure Spaces (2010, background): “ Finite or inﬁnite dimensional Gaussian spaces [Stu06a], [LV09]. Slightly modiﬁed versions are satisﬁed for • Inﬁnite dimensional spaces, like the Wiener space [FSS], as well as for • Discrete spaces [BS09], [Oll09]. Numerous important geometric and functional analytic estimates can be deduced from ”
Ricci curvature of Markov chains on metric spaces (2007, background): “Second, it is infinitesimal, and difficult to adapt to discrete settings [ BS ].”
Displacement convexity of entropy and related inequalities on graphs (2012, background): “However, while a lot is nowknown in the Riemannian setting (and more generally in geodesic spaces), very little is known so far in discrete spaces (such as finite graphs or finite Markov chains), with the notable exception of some notions of (discrete) Ricci curvature proposed recently by several au”
Gradient flows of the entropy for finite Markov chains (2011, background): “Furthermore, the Lott-Sturm-Villani definition of Ricci curvature [17, 26] cannot be applied in the discrete setting, sinceW2-geodesics between distinct elements of P(Q1) do not exist.”
Volume growth, spectrum and stochastic completeness of infinite graphs (2011, background): “Several other notions of curvature have been introduced for planar graphs [4, 5, 31], cell complexes [21], and general metric measure spaces [8, 43, 45, 49, 50].”
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”.”
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”