Strong solutions of stochastic equations with singular time dependent drift

We prove existence and uniqueness of strong solutions to stochastic equations in domains with unit diffusion and singular time dependent drift b up to an explosion time. We only assume local L_q_L_p-integrability of b in ×G with d/p+2/q<1. We also prove strong Feller properties in this case. If b is the gradient in x of a nonnegative function blowing up as GxG, we prove that the conditions 2D_tK,2D_t+Ke, [0,2), imply that the explosion time is infinite and the distributions of the solution have sub Gaussian tails.The work of the first author was partially supported by NSF Grant DMS-0140405Mathematics Subject Classification (2000): 60J60, 31C25Acknowledgement Financial support by the Humboldt Foundation, the BiBoS-Research Centre and the DFG-Forschergruppe Spectral Analysis, Asymptotic Distributions and Stochastic Dynamics is gratefully acknowledged. The authors are also sincerely grateful to the referees for their helpful suggestions.