Stochastic differential equations with critically irregular drift coefficients

Abstract

This paper is concerned with stochastic differential equations (SDEs for short) with irregular coefficients. By utilising a functional analytic approximation approach, we establish the existence and uniqueness of strong solutions to a class of SDEs with critically irregular drift coefficients in a new critical Lebesgue space, where the element may be only weakly integrable in time. To be more precise, let b:[0,T]×Rd→Rd be Borel measurable, where T>0 is arbitrarily fixed and d⩾1. We consider the following SDEXt=x+∫0tb(s,Xs)ds+Wt,t∈[0,T],x∈Rd, where {Wt}t∈[0,T] is a d-dimensional standard Wiener process. For p,q∈[1,+∞), we denote by C[q]([0,T];Lp(Rd)) the space of all Borel measurable functions f such that t1qf(t)∈C([0,T];Lp(Rd)). If b=b1+b2 such that |b1(T−⋅)|∈C[q]([0,T];Lp(Rd)) with 2/q+d/p=1 and ‖b1(T−⋅)‖C[q]([0,T];Lp(Rd)) is sufficiently small, and that b2 is bounded and Borel measurable, then we show that there exists a weak solution to the above equation, and if in addition limt↓0⁡‖t1qb(T−t)‖Lp(Rd)=0, the pathwise uniqueness holds. Furthermore, we obtain the strong Feller property of the semi-group and the existence of density associated with the above SDE. Besides, we extend the classical results concerning partial differential equations (PDEs) of parabolic type with Lq(0,T;Lp(Rd)) coefficients to the case of parabolic PDEs with L[q]∞(0,T;Lp(Rd)) coefficients, and derive the Lipschitz regularity for solutions of second order parabolic PDEs (see Theorem 3.1). Our results extend Krylov-Röckner and Krylov's profound results of SDEs with singular time dependent drift coefficients [20,23] to the critical case of SDEs with critically irregular drift coefficients in a new critical Lebesgue space.