SDEs with critical general distributional drifts: sharp solvability and blow-ups

TL;DR

This work develops weak well-posedness for stochastic differential equations with critical, highly singular distributional drifts and potentially discontinuous diffusion, by representing the drift as a sum of a form-bounded component and a divergence-free BMO^{-1} part. The analysis hinges on connecting weak solutions of the Kolmogorov backward PDE in the standard Hilbert triple to the stochastic dynamics, via De Giorgi’s method, Trotter's approximation, and compensated compactness. It yields dimension-free thresholds (δ<4) for well-posedness, extends to diffusion matrices with form-bounded gradients, and applies to finite-particle systems (e.g., Keller–Segel-type models) to derive improved bounds on the many-particle Hardy inequality, showing near-optimality relative to known lower bounds. The results sharpen the link between PDE regularity and stochastic dynamics in blow-up regimes and have concrete implications for interacting particle models in turbulent flows.

Abstract

We establish weak well-posedness for SDEs having discontinuous diffusion coefficients and general distributional drifts that may introduce local blow up effects. Our drifts satisfy minimal assumptions, i.e.\,we assume only that the Cauchy problem for the Kolmogorov backward equation is well-posed in the standard Hilbert triple $W^{1,2} \hookrightarrow L^2 \hookrightarrow W^{-1,2}$. By a result of Mazya and Verbitsky, these assumptions are precisely those drifts that can be represented as the sum of a form-bounded component (encompassing, for example, Morrey or Chang-Wilson-Wolff drifts) and a divergence-free distributional component in the ${\rm BMO}^{-1}$ space of Koch and Tataru. We apply our results to finite particle systems with strong attracting interactions immersed in a turbulent flow. This includes particle systems of Keller-Segel type. Crucially, in dimensions $d \geq 3$, we cover almost the entire admissible range of attraction strengths, reaching nearly to the blow-up threshold. As a further application of our results for SDEs and of the theory of Bessel processes, we obtain an improved upper bound on the constant in the many-particle Hardy inequality. Consequently, the lower bound previously derived by Hoffmann-Ostenhof, Hoffmann-Ostenhof, Laptev, and Tidblom is shown to be close to optimal.

Theorems & Definitions (50)

• Proposition 2.1: CLMS
• Proposition 2.2
• Lemma 2.1
• Lemma 3.1: Ki_multi
• Theorem 5.1
• Theorem 6.1
• Theorem 6.2: An upper bound on the constant in the many particle Hardy inequality \ref{['multi_hardy']}
• proof
• Theorem 6.3
• Theorem 7.1