Characterization of absolutely continuous curves in Wasserstein spaces (Q859772)

The author gives a characterization of absolutely continuous curves with finite p-energy in the Wasserstein space \(\mathcal{P}_p(X)\) of Borel probability measures with finite \(p\)-moments defined on a separable, complete metric space \(X\). In particular he proves that it is possible to represent them by Borel probability measures on the space \(C([0,T]; X)\) concentrated on the set of absolutely continuous curves with finite \(p\)-energy in \(X\). The author applies this result to give a characterization of the geodesics of \(\mathcal{P}_p(X)\) and to the continuity equation in Banach spaces.