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 clas-sical 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 direc-tions: we give a short proof of a result by Strassen on the existence of a martingale with given marginals, we characterize the associ-ated 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. 1