Existence and Uniqueness of Solutions to Fokker-Planck Type Equations with Irregular Coefficients

Claude Le Bris (1, 2) , Pierre Louis Lions (3, 4)

1 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique
2 MICMAC - Methods and engineering of multiscale computing from atom to continuum
3 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
4 Collège de France - Chaire Équations aux dérivées partielles et applications

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]. 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].

Domaines: Mathématiques générales [math.GM]

Claude Le Bris, Pierre Louis Lions. Existence and Uniqueness of Solutions to Fokker-Planck Type Equations with Irregular Coefficients. Communications in Partial Differential Equations, 2008, 33 (7), pp.1272-1317. ⟨10.1080/03605300801970952⟩. ⟨hal-00667315⟩