A Theory of Transfers: Duality and convolution

Abstract: We introduce and study the permanence properties of the class of linear transfers between probability measures. This class contains all cost minimizing mass transports, but also martingale mass transports, the Schrodinger bridge associated to a reversible Markov process, and the weak mass transports of Tala- grand, Marton, Gozlan and others. The class also includes various stochastic mass transports to which Monge-Kantorovich theory does not apply. We also introduce the cone of convex transfers, which include any p-power (p > 1) of a linear transfer, but also the logarithmic entropy, the Donsker-Varadhan infor- mation and certain free energy functionals. This first paper is mostly about exhibiting examples that point to the pervasiveness of the concept in the important work on correlating probability distributions. Duality formulae for general transfer inequalities follow in a very natural way. We also study the infinite self-convolution of a linear transfer in order to establish the existence of generalized weak KAM solutions that could be applied to the stochastic counterpart of Fathi-Mather theory.