The It\^o SDEs and Fokker--Planck equations with Osgood and Sobolev coefficients

Abstract

We study the degenerated It\^o SDE on $\mathbb R^d$ whose drift coefficient only fulfills a mixed Osgood and Sobolev regularity. Under suitable assumptions on the gradient of the diffusion coefficient and on the divergence of the drift coefficient, we prove the existence and uniqueness of generalized stochastic flows associated to such equations. We also prove the uniqueness of solutions to the corresponding Fokker--Planck equation by using the probabilistic method.