The Annals of Probability

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

Salah-Eldin A. Mohammed, Torstein K. Nilssen, and Frank N. Proske

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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

\[\cases{d_{t}u(t,x)+(b(t,x)\cdot Du(t,x))\,dt+\sum_{i=1}^{d}e_{i}\cdot Du(t,x)\circ dB_{t}^{i}=0,\cr u(0,x)=u_{0}(x),}\] 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.

Article information

Source
Ann. Probab., Volume 43, Number 3 (2015), 1535-1576.

Dates
First available in Project Euclid: 5 May 2015

Permanent link to this document
https://projecteuclid.org/euclid.aop/1430830289

Digital Object Identifier
doi:10.1214/14-AOP909

Mathematical Reviews number (MathSciNet)
MR3342670

Zentralblatt MATH identifier
1333.60127

Subjects
Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60H15: Stochastic partial differential equations [See also 35R60] 37H05: Foundations, general theory of cocycles, algebraic ergodic theory [See also 37Axx] 37H10: Generation, random and stochastic difference and differential equations [See also 34F05, 34K50, 60H10, 60H15] 34A36: Discontinuous equations

Keywords
SDEs with measurable coefficients stochastic flows Malliavin derivatives Sobolev spaces stochastic transport equation

Citation

Mohammed, Salah-Eldin A.; Nilssen, Torstein K.; Proske, Frank N. Sobolev differentiable stochastic flows for SDEs with singular coefficients: Applications to the transport equation. Ann. Probab. 43 (2015), no. 3, 1535--1576. doi:10.1214/14-AOP909. https://projecteuclid.org/euclid.aop/1430830289


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