  In this paper, we study the weak differentiability of global strong solution
of stochastic differential equations, the strong Feller property of the
associated diffusion semigroups and the global stochastic flow property in
which the singular drift $b$ and the weak gradient of Sobolev diffusion
$\sigma$ are supposed to satisfy $||b(x){1}_{|x|\le R}||_{p_1}\le O((\log
R)^{{(p_1-d)^2}/{2p^2_1}})$ and $||\nabla \sigma(x)1_{|x|\le R} ||_{p_1} \le
O((\log ({R}/{3}))^{{(p_1-d)^2}/{2p^2_1}})$ respectively. The main tools for
these results are the decomposition of global two-point motions, Krylov's
estimate, Khasminskii's estimate, Zvonkin's transformation and the
characterization for Sobolev differentiability of random fields.
