Stochastic differential equations with critically irregular drift coefficients - Jiang-lun Wu (Journal of Differential Equations, 2023)
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→Rdbe Borel measurable, where T>0is arbitrarily fixed and d⩾1. We consider the following SDE Xt=x+ t 0 b(s,Xs)ds+Wt,t∈[0,T],x∈Rd, where Wtt∈[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 fsuch that t1 qf(t) ∈C([0, T]; Lp(Rd)). If b=b1+b2such that |b1(T−·)| ∈C[q]([0, T]; Lp(Rd))with 2/q+d/p=1and b1(T− ·)C[q]([0,T];Lp(Rd))is sufficiently small, and that b2is bounded and Borel measurable, then we show that there exists a weak solution to the above equation, and if in addition limt↓0t1 qb(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 Theorem3.1). 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.