Economic couplings and acyclic flows

To any coupling between two probability measures $\mu_1$ and $\mu_2$ on a finite set there is naturally associated a flow that represents the amount of mass transported to change the measure $\mu_1$ into the measure $\mu_2$. This flow is not unique since several different choices of paths are possible. We prove that also a converse statement holds when the flow is acyclic. In particular we show two different algorithms that associate to any acyclic flow having divergence coinciding with $\mu_1-\mu_2$ a coupling between the two measures. The couplings that can be obtained in this way are called \emph{economic}. In the case of a countable set the constructions are implemented with a limit procedure and the flows for which they hold need to satisfy a suitable \emph{finite decomposability} condition. We show several consequences of these constructive procedures among which a third equivalent statement in Strassen Theorem and a constructive proof of the equivalence between a mass transportation problem with a geodesic cost and a minimal current problem. We illustrate the results discussing several solvable cases