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

TL;DR

This paper establishes weak well-posedness of energy solutions for singular stochastic differential equations with supercritical distributional drift, extending the understanding of such equations in high-dimensional and non-divergence free cases.

Contribution

It introduces new weak well-posedness results for energy solutions of singular SDEs with supercritical distributional drift, including divergence-free and certain non-divergence free cases.

Findings

Weak well-posedness proven for time-dependent divergence-free drifts.

Weak well-posedness established for certain time-independent drifts with local singularities.

Results apply in high dimensions with drifts outside classical Besov space classes.

Abstract

We study stochastic differential equations with additive noise and distributional drift on $\mathbb{T}^d$ or $\mathbb{R}^d$ and $d \geqslant 2$. 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 $b \in L^p_T B^{-\gamma}_{p, 1}$ with $p \in (2, \infty]$ and $p \geqslant \frac{2}{1 -\gamma}$. 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 $b \notin B^{-1}_{2 d/(d-2), 2}$, meaning that for any $p > 2$ in sufficiently high dimension there exists $b \notin B^{-1}_{p, 2}$ such that weak well-posedness holds for energy solutions with drift $b$.