Speaker
Description
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 Savaré (Milano).
