Stochastic Differential Equations with Local Growth Singular Drifts

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 \(\left\| \left| b\right| \cdot \mathbbm {1}_{B(R)}\right\| _{p_1}\le O((\log R)^{{(p_1-d)^2}/{2p^2_1}})\) and \(\left\| \left\| \nabla \sigma \right\| \cdot \mathbbm {1}_{B(R)}\right\| _{p_1}\le O((\log ({R}/{3}))^{{(p_1-d)^2}/{2p^2_1}})\), respectively. The main tools for these results are the decomposition of global two-point motions in Fang et al. (Ann Probab 35(1):180–205, 2007), Krylov’s estimate, Khasminskii’s estimate, Zvonkin’s transformation and the characterization for Sobolev differentiability of random fields in Xie and Zhang (Ann Probab 44(6):3661–3687, 2016).