SDEs with critical time dependent drifts: Weak solutions

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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,∞) and divb∈L∞([0,T];Ld∕2+ε(Rd)), 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.