Stochastic differential equations with critical drifts

Abstract

We establish the well-posedness of SDE with the additive noise when a singular drift belongs to the critical spaces. We prove that if the drift belongs to the Orlicz-critical space L-q,L-1([0, T], L-x(p)) for p, q is an element of (1, infinity) satisfying 2/q + d/p =1, then the corresponding SDE admits a unique strong solution. We also derive the Sobolev regularity of a solution under the Orlicz-critical condition. (C) 2020 Elsevier B.V. All rights reserved.