Semiclassical limit of polaron dynamics with cutoff

We define and study the global well-posedness of a classical polaron dynamics and its derivation from the quantum dynamics of the Fröhlich Hamiltonian from the viewpoint of Wigner measures. We make a crucial use of the Gross transform, allowing us to extract a diverging constant. Finally, we discuss the difficulty of defining the classical dynamics when removing the cutoff in our setting.