Stochastic differential equations with coefficients in Sobolev spaces

We consider the Itô stochastic differential equation on Rd. The diffusion coefficients A1,…,Am are supposed to be in the Sobolev space with p>d, and to have linear growth. For the drift coefficient A0, we distinguish two cases: (i) A0 is a continuous vector field whose distributional divergence δ(A0) with respect to the Gaussian measure γd exists, (ii) A0 has Sobolev regularity for some p′>1. Assume for some λ0>0. In case (i), if the pathwise uniqueness of solutions holds, then the push-forward (Xt)#γd admits a density with respect to γd. In particular, if the coefficients are bounded Lipschitz continuous, then Xt leaves the Lebesgue measure Lebd quasi-invariant. In case (ii), we develop a method used by G. Crippa and C. De Lellis for ODE and implemented by X. Zhang for SDE, to establish existence and uniqueness of stochastic flow of maps.