Strong solutions of stochastic equations with singular time dependent drift

Abstract

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 ℝ×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 t ψ≤Kψ,2D t ψ+Δψ≤Ke ɛψ ,ɛ ∈ [0,2), imply that the explosion time is infinite and the distributions of the solution have sub Gaussian tails.