  In this paper, we establish the existence of a stochastic flow of Sobolev
diffeomorphisms \[\mathbb{R}^d\ni x\quad\longmapsto\quad\phi_{s,t}(x)\in
\mathbb{R}^d,\qquad s,t\in\mathbb{R}\] for a stochastic differential equation
(SDE) of the form \[dX_t=b(t,X_t)\,dt+dB_t,\qquad
s,t\in\mathbb{R},X_s=x\in\mathbb{R}^d.\] The above SDE is driven by a bounded
measurable drift coefficient
$b:\mathbb{R}\times\mathbb{R}^d\rightarrow\mathbb{R}^d$ and a $d$-dimensional
Brownian motion $B$. More specifically, we show that the stochastic flow
$\phi_{s,t}(\cdot)$ of the SDE lives in the space
$L^2(\Omega;W^{1,p}(\mathbb{R}^d,w))$ for all $s,t$ and all $p\in (1,\infty)$,
where $W^{1,p}(\mathbb{R}^d,w)$ denotes a weighted Sobolev space with weight
$w$ possessing a $p$th moment with respect to Lebesgue measure on $\mathbb
{R}^d$. From the viewpoint of stochastic (and deterministic) dynamical systems,
this is a striking result, since the dominant "culture" in these dynamical
systems is that the flow "inherits" its spatial regularity from that of the
driving vector fields. The spatial regularity of the stochastic flow yields
existence and uniqueness of a Sobolev differentiable weak solution of the
(Stratonovich) stochastic transport equation \[\cases{\displaystyle
d_tu(t,x)+\bigl(b(t,x)\cdot Du(t,x)\bigr)\,dt+\sum_{i=1}^de_i\cdot Du(t,x)\circ
dB_t^i=0,\cr u(0,x)=u_0(x),}\] where $b$ is bounded and measurable, $u_0$ is
$C_b^1$ and $\{e_i\}_{i=1}^d$ a basis for $\mathbb{R}^d$. It is well known that
the deterministic counterpart of the above equation does not in general have a
solution.

Probability