Energy solutions to SDEs with supercritical distributional drift: A stopping argument

In this note we consider the SDE {equation*} dX_t = b (t, X_t) d t + √2 d B_t, {mainSDE} {equation*} in dimension d 2, where B is a Brownian motion and b : R₊ → S' (Rᵈ ; Rᵈ) is distributional, scaling super-critical and satisfies ∇ · b ≡ 0. We partially extend the super-critical weak well-posedness result for energy solutions from [GP24] by allowing a mixture of the regularity regimes treated therein: Outside of environments of a small (and compared to [GP24] ''time-dependent'') set K ⊂ R₊ × Rᵈ, the antisymmetric matrix field A with bⁱ = ∇· Aᵢ is assumed to be in a certain supercritical LqT Hs, p-type class that allows a direct link between the PDE and the SDE from a-priori estimates up to the stopping time of visiting K. To establish this correspondence, and thus uniqueness, globally in time we then show that K is actually never visited which requires us to impose a relation between the dimension of K and the H\"older regularity of X.