SDEs with supercritical distributional drifts

Abstract

Let $d\ge 2$. In this paper, we investigate the following stochastic differential equation (SDE) in ${\mathbb{R}}^d$ driven by Brownian motion $$\mathrm{d}{X}_t=b(t,X_t)\mathrm{d}t+\sqrt{2}\mathrm{d}{W}_t,$$ where $b$ belongs to the space ${\mathbb{L}}_T^q{\mathbf{H}}_p^{\hat{I}}\pm$ with $\hat{I}\pm \in [-1,0]$ and $p,q\in [2,\infty ]$, which is a distribution-valued and divergence-free vector field. In the subcritical case $\frac{d}{p}+\frac{2}{q}<1+\hat{I}\pm$, we establish the existence and uniqueness of a weak solution to the integral equation: $$X_t=X_0+\underset{n\to \infty }{\lim }{\int }_0^t{b}_n(s,X_s)\mathrm{d}s+\sqrt{2}{W}_t.$$ Here, $b_n:=b\ast \ddot{I}{•}_n$ represents the mollifying approximation, and the limit is taken in the $L^2$-sense. In the critical and supercritical case $1+\hat{I}\pm \le \frac{d}{p}+\frac{2}{q}<2+\hat{I}\pm$, 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 $b=b_1+b_2+\mathrm{div}a$, where $b_1\in {\mathbb{L}}_T^{\infty }{\mathbf{B}}_{\infty ,2}^{-1}$, $b_2\in {\mathbb{L}}_T^2{L}^2$, 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.

40pages