Kantorovich duality for general transport costs and applications

Abstract

We introduce a general notion of transport cost that encompasses many costs used in the literature (including the classical one and weak transport costs introduced by Talagrand and Marton in the 90's), and prove a Kantorovich type duality theorem. As a by-product we obtain various applications in different directions: we give a short proof of a result by Strassen on the existence of a martingale with given marginals, we characterize the associated transport-entropy inequalities together with the log-Sobolev inequality restricted to convex/concave functions. Some explicit examples of discrete measures satisfying weak transport-entropy inequalities are also given. December 25, 2015. 1991 Mathematics Subject Classification. 60E15, 32F32 and 26D10. Date:

Concentration of Measure Principle and Entropy-Inequalities

“…The so-called barycentric costs are defined on X = R n equipped with the Euclidean distance, d(x, y) = |x − y|, x, y ∈ R n . They have been introduced in the paper [GRST14b] to reach optimal concentration properties for discrete measures (see Section 4.2). As explained in [GRST14b], they are also related to the convex (τ )-property by Maurey [Mau91].…”

“…A functional formulation of the concentration principle of Definition 2.1 is presented in [GRST14b]. This second definition is associated to the following type of infimum-convolution operator, introduced in [Sam07,GRST14b]: for any mesurable function ϕ : X → R ∪ {∞} bounded from below…”

“…This second definition is associated to the following type of infimum-convolution operator, introduced in [Sam07,GRST14b]: for any mesurable function ϕ : X → R ∪ {∞} bounded from below…”

“…(see Lemma 5.6 [GRST14b]). This inclusion depends of the kind of enlargement and is not satisfied for some enlargements.…”

“…The definition of this principle with enlargements of sets takes its origin from the papers by M. Talagrand [Tal95,Tal96b,Tal96a]. We propose a functional formulation of the concentration principle, rigorously introduced in [GRST14b]. We emphasize three types of cost functions that provide most of the enlargements of sets considered in the literature, the usual cost functions, the barycentric cost functions and the universal cost functions.…”