Stochastic differential equations with coefficients in Sobolev spaces

We consider the Itô stochastic differential equation dXt = m j=1 Aj (Xt ) dw j t + A0(Xt ) dt on Rd. The diffusion coefficients A1, . . . , Am are supposed to be in the Sobolev space W 1,p loc (Rd ) 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 W 1,p loc for some p > 1. Assume Rd exp[λ0(|δ(A0)|+ m j=1(|δ(Aj )|2 +|∇Aj |2))]dγd <+∞ 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. © 2010 Elsevier Inc. All rights reserved.