Curvature and transport inequalities for Markov chains in discrete spaces

By Max Fathi, Y. Shu. 2015

54 citations recorded by the source corpus.

We study various transport-information inequalities under three different notions of Ricci curvature in the discrete setting: the curvature-dimension condition of Bakry and \'Emery, the exponential curvature-dimension condition of Bauer \textit{et al.} and the coarse Ricci curvature of Ollivier. We prove that under a curvature-dimension condition or coarse Ricci curvature condition, an $L_1$ transport-information inequality holds; while under an exponential curvature-dimension condition, some weak-transport information inequalities hold. As an application, we establish a Bonnet-Meyer's theorem under the curvature-dimension condition CD$(\kappa,\infty)$ of Bakry and \'Emery.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/e4a5c19bb3943c98f63168277390110bd5e88c38

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Geometry and Topology

Citations in the corpus

On a mixture of Brenier and Strassen Theorems (2018, methodology): “We refer to [30, 29, 64, 65, 20] for works directly connected to [31] and to [63] for an up-to-date survey of applications of weak transport costs to concentration of measure.”
Bakry–Émery curvature and diameter bounds on graphs (2016, background): “K(x, y) := { w(x,y) m(x) Degmax : x 6= y 1− Deg(x) Degmax : x = y and the corresponding measure μ(x) := m(x)/m(V ) for all x ∈ V according to [7].”
Applications of weak transport theory (2020, unknown): “Overview The optimal transport problem for weak costs was first introduced by Gozlan, Roberto, Samson and Tetali [34] and has immediately generated interest in several groups of researchers, see [45, 46, 29, 32, 1, 2] among others.”
Existence, duality, and cyclical monotonicity for weak transport costs (2018, background): “Further papers directly related to [25] include [37, 36, 38, 21, 23].”