On weak solutions of stochastic differential equations with sharp drift coefficients

Abstract

We extend Krylov and Röckner's result \cite{KR} to the drift coefficients in critical Lebesgue space, and prove the existence and uniqueness of weak solutions for a class of SDEs. To be more precise, let $b:[0,T]\times {\mathbb{R}}^d\to {\mathbb{R}}^d$ be Borel measurable, where $T>0$ is arbitrarily fixed. Consider $$X_t=x+{\int }_0^{t}b(s,X_s)ds+W_t,t\in [0,T],x\in {\mathbb{R}}^d,$$ where $\{W_t{\}}_{t\in [0,T]}$ is a $d$-dimensional standard Wiener process. If $b=b_1+b_2$ such that $b_1(T-\cdot )\in {\mathcal{C}}_q^0((0,T];L^p({\mathbb{R}}^d))$ with $2/q+d/p=1$ for $p,q\ge 1$ and $\Vert b_1(T-\cdot ){\Vert }_{{\mathcal{C}}_q((0,T];L^p({\mathbb{R}}^d))}$ is sufficiently small, and that $b_2$ is bounded and Borel measurable, then there exits a unique weak solution to the above equation. Furthermore, we obtain the strong Feller property of the semi-group and existence of density associated with above SDE. Besides, we extend the classical partial differential equations (PDEs) results for $L^q(0,T;L^p({\mathbb{R}}^d))$ coefficients to $L_q^{\infty }(0,T;L^p({\mathbb{R}}^d))$ ones, and derive the Lipschitz regularity for solutions of second order parabolic PDEs (see Lemma 2.1).