Sobolev differentiable stochastic flows for SDEs with singular coefficients: Applications to the transport equation

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.