ON GLOBAL EXISTENCE OF SOLUTIONS OF SDES WITH SINGULAR DRIFT

Let b be a Borel measurable IR(d)-valued function, defined on some Bor el subset of IR(d) Consider the d-dimensional SDE X(t) = X(o) + integr al(o)(t) b(X(s)) ds + W-t with singular drift b. A local solution (up to sigma) is a tuple (X, W, Q, sigma) where X is a stochastic process, W is a Brownian motion under the probability measure Q, and sigma is a strictly optional time (i.e., stopping time) such that the above equ ation is satisfied for all t < sigma. Such a local solution was constr ucted 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 solu tion, i.e., for Q(sigma = infinity) = 1.