Strong solutions of stochastic equations with singular time dependent drift

We prove existence and uniqueness of strong solutions to stochastic equations in domains G ⊂ ℝ^d 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 ℝ x 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 ψ blowing up as G ∋ x → ∂G, we prove that the conditions 2D_tψ ≤ Kψ, 2D_tψ + Δψ ≤ Ke^εψ, ε ∈ [0, 2), imply that the explosion time is infinite and the distributions of the solution have sub Gaussian tails.