Weak well-posedness of energy solutions to singular SDEs with supercritical distributional drift

We study stochastic differential equations with additive noise and distributional drift on ${\mathbb{T}}^d$ or ${\mathbb{R}}^d$ and $d⩾2$. We work in a scaling-supercritical regime using energy solutions and recent ideas for generators of singular stochastic partial differential equations. We mainly focus on divergence-free drift, but allow for scaling-critical non-divergence free perturbations. In the time-dependent divergence-free case we roughly speaking prove weak well-posedness of energy solutions with initial law $\mu \ll \text{Leb}$ for drift $b\in L_T^p{B}_{p,1}^{-\gamma }$ with $p\in (2,\mathrm{\infty }]$ and $p⩾\frac{2}{1-\gamma }$. For time-independent $b$ we show weak well-posedness of energy solutions with initial law $\mu \ll \text{Leb}$ under certain structural assumptions on $b$ which allow local singularities such that $b\mathrm{\notin }{B}_{2d\mathrm{/}(d-2),2}^{-1}$, meaning that for any $p>2$ in sufficiently high dimension there exists $b\mathrm{\notin }{B}_{p,2}^{-1}$ such that weak well-posedness holds for energy solutions with drift $b$.