SDEs with critical time dependent drifts: weak solutions

We prove the unique weak solvability of time-inhomogeneous stochastic differential equations with additive noises and drifts in critical Lebsgue space $L^q([0,T];L^p({\mathbb{R}}^d))$ with $d\mathrm{/}p+2\mathrm{/}q=1$. The weak uniqueness is obtained by solving corresponding Kolmogorov’s backward equations in some second order Sobolev spaces, which is analytically interesting in itself.