  The goal of this paper is to study weak solutions of the Fokker-Planck
equation. We first discuss existence and uniqueness of weak solutions in an
irregular context, providing a unified treatment of the available literature
along with some extensions. Then, we prove a regularity result for
distributional solutions under suitable integrability assumptions, relying on a
new, simple commutator estimate in the spirit of DiPerna-Lions' theory of
renormalized solutions for the transport equation. Our result is somehow
transverse to Theorem 4.3 of [15]: on the diffusion matrix we relax the
assumption of Lipschitz regularity in time at the price of assuming Sobolev
regularity in space, and we prove the regularity (and hence the uniqueness) of
distributional solutions to the Fokker-Planck equation.
