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 μ Leb for drift b ∈ LᵖT B-γp, 1 with p ∈ (2, ∞] and p 2/1 -γ. For time-independent b we show weak well-posedness of energy solutions with initial law μ Leb under certain structural assumptions on b which allow local singularities such that b ∉ B⁻¹2 d/(d-2), 2, meaning that for any $p > 2$ in sufficiently high dimension there exists b ∉ B⁻¹p, 2 such that weak well-posedness holds for energy solutions with drift b.