Existence and uniqueness of solutions to Fokker-Planck type equations with irregular coefficients

We study the existence and the uniqueness of the solution to a class of Fokker-Planck type equations with irregular coefficients, more precisely with coefficients in Sobolev spaces W-1,W- p. Our arguments are based upon the DiPerna-Lions theory of renormalized solutions to linear transport equations and related equations [5]. The present work extends the results of our previous article [14], where only the simpler case of a Fokker-Planck equation with constant diffusion matrix was addressed. The consequences of the present results on the well-posedness of the associated stochastic differential equations are only outlined here. They will be more thoroughly examined in a forthcoming work [15].