$z$. More precisely, for any $f\in C^2_c(\mR^{2d})$, $$\p_t\mu_t(f)=\mu_t(\sL^{a,b}_t f),\ \mu_0(f)=f(z),$$ where $\mu_t(f)=\int f\dif\mu_t=\mE f(Z_t)$. In the literature $\sL^{a,b}_t$ is also called kinetic Fokker-Planck or Kolmogorov’s operator.

During the past decade, there is an increasing interest in the study of SDEs with singular or rough coefficients. In the non-degenerate case, Krylov and Röckner [@Kr-Ro] showed the strong uniqueness to the following SDE in $\mR^d$: $$\dif X_t=b_t(X_t)\dif t+\dif W_t, \ X_0=x,$$ where $b\in L^q_{lock}(\mR^+;L^p(\mR^d))$ with $\frac{d}{p}+\frac{2}{q}<1$. The argument in [@Kr-Ro] is based on Girsanov’s theorem and some estimates from the theory of PDE. In this framework, Fedrizzi and Flandoli [@Fe-Fl; @Fe-Fl0] studied the well-posedness of stochastic transport equations with rough coefficients. When $b$ is bounded measurable, the Malliavin differentiability of $X_t$ with respect to sample path $\omega$ and the weak differentiability of $X_t$ with respect to starting point $x$ were recently studied in [@Me] and [@Mo-Ni-Pr] respectively. We also mention that weak uniqueness was studied in [@Ba-Ch] and [@Ji] under rather weak assumptions on $b$ (belonging to some Kato’s class). Moreover, the multiplicative noise case was studied in [@Zh0; @Zh2; @Zh4] by using Zvonkin’s transformation [@Zv] and some careful estimates of second order parabolic equations.

In the degenerate case, Chaudru de Raynal 