We prove existence and uniqueness of strong solutions to stochastic equations in domains G ⊂ ℝd with unit diffusion and singular time dependent drift b up to an explosion time. We only assume local L q_Lp-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 2Dtψ ≤ Kψ, 2Dtψ + Δψ ≤ Keεψ, ε ∈ [0, 2), imply that the explosion time is infinite and the distributions of the solution have sub Gaussian tails.
- Distorted Brownian motion
- Singular drift
- Strong solutions of stochastic equations