Kantorovich duality for general transport costs and applications

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...

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Bibliographic Details
Published inJournal of functional analysis Vol. 273; no. 11; pp. 3327 - 3405
Main Authors Gozlan, Nathael, Roberto, Cyril, Samson, Paul-Marie, Tetali, Prasad
Format Journal Article
LanguageEnglish
Published Elsevier Inc 01.12.2017
Subjects
Duality
Logarithmic-Sobolev inequalities
Metric spaces
Transport inequalities
60E15
32F32
Transport inequalities
26D10
Duality
Metric spaces
Logarithmic-Sobolev inequalities
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Summary: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. We also provide explicit examples of discrete measures satisfying the weak transport-entropy inequalities derived here.
ISSN:0022-1236
1096-0783
DOI:10.1016/j.jfa.2017.08.015