Characterization of absolutely continuous curves in Wasserstein spaces

Let X be a separable, complete metric space and Pp(X) be the space of Borel probability measures with finite moment of order p > 1, metrized by the Wasserstein distance. In this paper we prove that every absolutely continuous curve with finite p-energy in the space Pp(X) can be represented by a Borel probability measure on C([0,T];X) concentrated on the set of absolutely continuous curves with finite p-energy in X. Moreover this measure satisfies a suitable property of minimality which entails an important relation on the energy of the curves. We apply this result to the geodesics of Pp(X) and to the continuity equation in Banach spaces.