SDEs with critical time dependent drifts: Weak solutions

For d≥3, we prove that time-inhomogeneous stochastic differential equations driven by additive noises with drifts in critical Lebesgue space Lq([0,T];Lp(Rd)), where (p,q)∈(d,∞]×[2,∞) and d∕p+2∕q=1, or (p,q)=(d,∞) divb∈L∞([0,T];Ld∕2+ε(Rd)), are well-posed. The weak uniqueness is obtained solving corresponding Kolmogorov backward some second-order Sobolev spaces, which analytically interesting itself.