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

TL;DR

This paper proves weak well-posedness for stochastic differential equations with discontinuous diffusion and complex distributional drifts, and applies these results to particle systems with strong interactions, including near blow-up scenarios.

Contribution

It establishes well-posedness under minimal assumptions on drifts, extending the theory to include form-bounded and divergence-free components, and applies findings to particle systems and Hardy inequalities.

Findings

Proved weak well-posedness for SDEs with critical distributional drifts.

Extended applicability to particle systems with strong attraction in high dimensions.

Improved bounds on constants in many-particle Hardy inequalities.

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.