SDEs with supercritical distributional drifts

TL;DR

This paper studies SDEs with distributional, divergence-free drifts in supercritical regimes, establishing existence, uniqueness, and convergence of solutions, and providing examples like vortex models and Gaussian fields.

Contribution

It extends the theory of SDEs to supercritical distributional drifts, proving existence and convergence results in new regimes with minimal assumptions.

Findings

Existence and uniqueness of weak solutions in subcritical case.

Existence of solutions in supercritical case with initial density.

Convergence of mollified solutions under specific structural assumptions.

Abstract

Let $d\geq 2$. In this paper, we investigate the following stochastic differential equation (SDE) in ${\mathbb R}^d$ driven by Brownian motion $$ {\rm d} X_t=b(t,X_t){\rm d} t+\sqrt{2}{\rm d} W_t, $$ where $b$ belongs to the space ${\mathbb L}_T^q \mathbf{H}_p^\alpha$ with $\alpha \in [-1, 0]$ and $p,q\in[2, \infty]$, which is a distribution-valued and divergence-free vector field. In the subcritical case $\frac dp+\frac 2q<1+\alpha$, we establish the existence and uniqueness of a weak solution to the integral equation: $$ X_t=X_0+\lim_{n\to\infty}\int^t_0b_n(s,X_s){\rm d} s+\sqrt{2} W_t. $$ Here, $b_n:=b*\phi_n$ represents the mollifying approximation, and the limit is taken in the $L^2$-sense. In the critical and supercritical case $1+\alpha\leq\frac dp+\frac 2q<2+\alpha$, assuming the initial distribution has an $L^2$-density, we show the existence of weak solutions and associated Markov processes. Moreover, under the additional assumption that $b=b_1+b_2+{\rm div} a$, where $b_1\in {\mathbb L}^\infty_T{\mathbf B}^{-1}_{\infty,2}$, $b_2\in {\mathbb L}^2_TL^2$, and $a$ is a bounded antisymmetric matrix-valued function, we establish the convergence of mollifying approximation solutions without the need to subtract a subsequence. To illustrate our results, we provide examples of Gaussian random fields and singular interacting particle systems, including the two-dimensional vortex models.