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

TL;DR

This paper proves the existence of Sobolev differentiable stochastic flows for SDEs with singular, bounded measurable coefficients, enabling the construction of weak solutions to the associated stochastic transport equation.

Contribution

It establishes the Sobolev differentiability of stochastic flows for SDEs with irregular coefficients, a novel result in stochastic dynamical systems theory.

Findings

Stochastic flows are in weighted Sobolev spaces with integrability properties.

Existence and uniqueness of Sobolev weak solutions to the stochastic transport equation.

Flow regularity is achieved despite singular drift coefficients.

Abstract

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.