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, p . Our arguments are based upon the DiPerna–Lions theory of renormalized solutions to linear transport equations and related equations [5 Di Perna , R. J. , Lions , P. L. ( 1989 ). Ordinary differential equations, transport theory and Sobolev spaces . Invent. Math. 98 ( 3 ): 511 – 547 .[Crossref] , [Google Scholar]]. The present work extends the results of our previous article [14 Le Bris , C. , Lions , P. L. ( 2004 ). Renormalized solutions of some transport equations with partially W 1,1 velocities and applications . Annali di Matematica Pura ed Applicata 183 : 97 – 130 .[Crossref] , [Google Scholar]], 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 Le Bris , C. , Lions , P. L. Generalized flows for stochastic differential equations with irregular coefficients. In preparation . [Google Scholar]].