  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 almost sure 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.
