  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.
