Stochastic differential equations with coefficients in Sobolev spaces

We consider It^o SDE $\text{d}{X}_t={\sum }_{j=1}^m{A}_j(X_t)\text{d}{w}_t^j+A_0(X_t)\text{d}t$ on ${\mathbb{R}}^d$. The diffusion coefficients $A_1,\mathrm{.}\mathrm{.}\mathrm{.},A_m$ are supposed to be in the Sobolev space $W_{\text{loc}}^{1,p}({\mathbb{R}}^d)$ with $p>d$, and to have linear growth; for the drift coefficient $A_0$, we consider two cases: (i) $A_0$ is continuous whose distributional divergence $\delta (A_0)$ w.r.t. the Gaussian measure ${\gamma }_d$ exists, (ii) $A_0$ has the Sobolev regularity $W_{\text{loc}}^{1,p^{\mathrm{\prime }}}$ for some $p^{\mathrm{\prime }}>1$. Assume $\int_{\R^d} \exp\big[\lambda_0\bigl(|\delta(A_0)| + \sum_{j=1}^m (|\delta(A_j)|^2 +|\nabla A_j|^2)\bigr)\big] \d\gamma_d<+\infty$forsome$\lambda_0>0$, in the case (i), if the pathwise uniqueness of solutions holds, then the push-forward $(X_t)_# \gamma_d$admitsadensitywithrespectto$\gamma_d$. In particular, if the coefficients are bounded Lipschitz continuous, then $X_t$ leaves the Lebesgue measure ${\text{Leb}}_d$ quasi-invariant. In the 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 the existence and uniqueness of stochastic flow of maps.