Three superposition principles: Currents, continuity equations and curves of measures

We establish a general superposition principle for curves of measures solving a continuity equation on metric spaces without any smooth structure nor underlying measure, representing them as marginals of measures concentrated on the solutions of the associated ODE defined by some algebra of observables. We relate this result with decomposition of acyclic normal currents in metric spaces. As an application, a slightly extended version of a probabilistic representation for absolutely continuous curves in Kantorovich–Wasserstein spaces, originally due to S. Lisini, is provided in the metric framework. This gives a hierarchy of implications between superposition principles for curves of measures and for metric currents.