On Global Existence of Solutions of SDE's with Singular Drift

Let b be a Borel measurable IR^d - valued function, defined on some Borel subset of IR^d. Consider the d- dimensional SDE with singular drift b. A local solution (up to σ) is a tuple (X, W, Q,σ) where X is a stochastic process, W is a Brownian motion under the probability measure Q, and σ is a strictly optional time (i.e., stopping time) such that the above equation is satisfied for all t < σ. Such a local solution was constructed by the author in an earlier paper under very mild conditions on b. In this paper we give criteria for the global existence of the solution, i. e., for Q(σ = ∞) = 1.