Kantorovich duality for general transport costs and applications

TL;DR

This paper introduces a broad framework for transport costs, proves a Kantorovich duality theorem, and applies it to martingale existence, transport-entropy inequalities, and explicit examples with discrete measures.

Contribution

It generalizes transport costs, establishes a Kantorovich duality, and provides new applications including martingale existence and inequalities for discrete measures.

Findings

Unified framework for various transport costs

Kantorovich duality theorem proven for general costs

Explicit examples of discrete measures satisfying inequalities

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.