Representation results for Lipschitz (or even absolutely continuous) curves $\mu:[0,T]\to \mathcal P_p(\mathbb{R}^d)$, $p>1$, with values in the Wasserstein space $(\mathcal P_p(\mathbb{R}^d),W_p)$ of Borel probability measures in $\mathbb{R}^d$ with finite $p$-moment provide a crucial tool to study evolutionary PDEs and geometric problems in a measure-theoretic setting. They are strictly related to corresponding representation results for measure-valued solutions to the continuity equation, as a superposition of absolutely continuous curves solving a suitable differential equation. In this talk we discuss the validity and the appropriate formulation of the above results in the case $p=1$, for the space of probability measures with finite moment $\mathcal P_1(\mathbb{R}^d)$ endowed with the metric $W_1.$ We will thus provide a suitable version of the superposition principle for curves of measures in $\mathcal P_1(\mathbb{R}^d)$ that are only of  bounded variation with respect to the time variable.

Joint work with Stefano Almi (Napoli) and Giuseppe Savar\'e (Milano).