SDEs with critical time dependent drifts: Weak solutions

Abstract

For $d\ge 3$, we prove that time-inhomogeneous stochastic differential equations driven by additive noises with drifts in critical Lebesgue space $L^q([0,T];L^p({\mathbb{R}}^d))$, where $(p,q)\in (d,\mathrm{\infty }]\times [2,\mathrm{\infty })$ and $d∕p+2∕q=1$, or $(p,q)=(d,\mathrm{\infty })$ and $divb\in L^{\mathrm{\infty }}([0,T];L^{d∕2+\mathit{\epsilon }}({\mathbb{R}}^d))$, are well-posed. The weak uniqueness is obtained by solving corresponding Kolmogorov backward equations in some second-order Sobolev spaces, which is analytically interesting in itself.