Strong solutions of stochastic equations with singular time dependent drift (Q1769077)

Existence and uniqueness of solutions to stochastic differential equations in domains \(G \subset {\mathbb R}^d\) with unit local diffusion and singular time dependent drift \(b\) up to an explosion time are proved under the assumption of local \(L_{q^-}L_p \equiv L_q({\mathbb R}, L_p)\)-integrability on \(b\) in \({\mathbb R} \times G\) with \(d/p + 2/q < 1\). Strong Feller properties are also proved for this case. In the case when \(b\) is the gradient in \(x\) of a nonnegative function \(\psi\) blowing up as \(G \ni x\to \partial G\), it is proved that the conditions \(2D_t \psi \leq K \psi\), \(2 D_t \psi + \Delta \psi \leq K e^{\varepsilon \psi}\), \(\varepsilon \in [0, 2)\), imply that the explosion time is infinite and the distributions of the solution have sub-Gaussian tails.