Semiclassical limit of quantum dynamics with rough potentials and well posedness of transport equations with measure initial data

In this paper we study the semiclassical limit of the Schr\"odinger equation. Under mild regularity assumptions on the potential $U$ which include Born-Oppenheimer potential energy surfaces in molecular dynamics, we establish asymptotic validity of classical dynamics globally in space and time for "almost all" initial data, with respect to an appropriate reference measure on the space of initial data. In order to achieve this goal we prove existence, uniqueness and stability results for the flow in the space of measures induced by the continuity equation.