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 Tᵈ or Rᵈ 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 Leb for drift b LᵖT B^-₏, ₁ with p (2, ] and p 21 -. For time-independent b we show weak well-posedness of energy solutions with initial law Leb under certain structural assumptions on b which allow local singularities such that b B^-1₂ ₃/ (₃-₂), ₂, meaning that for any p > 2 in sufficiently high dimension there exists b B^-1₏, ₂ such that weak well-posedness holds for energy solutions with drift b.