Strong solutions of stochastic equations with singular time dependent drift

Krylov NV, Röckner M. Strong solutions of stochastic equations with singular time dependent drift. Probability Theory and Related Fields. 2005;131(2):154-196.We prove existence and uniqueness of strong solutions to stochastic equations in domains G subset of 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 partial derivativeG, we prove that the conditions 2D(t) psi less than or equal to Kpsi, 2D(t)psi + Deltapsi less than or equal to Ke(epsilonpsi), epsilon is an element of [0, 2), imply that the explosion time is infinite and the distributions of the solution have sub Gaussian tails