The Annals of Applied Probability

Stochastic differential equations with Sobolev diffusion and singular drift and applications

Xicheng Zhang

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Abstract

In this paper, we study properties of solutions to stochastic differential equations with Sobolev diffusion coefficients and singular drifts. The properties we study include stability with respect to the coefficients, weak differentiability with respect to starting points and the Malliavin differentiability with respect to sample paths. We also establish Bismut–Elworthy–Li’s formula for the solutions. As an application, we use the stochastic Lagrangian representation of incompressible Navier–Stokes equations given by Constantin–Iyer [Comm. Pure Appl. Math. 61 (2008) 330–345] to prove the local well-posedness of NSEs in $\mathbb{R}^{d}$ with initial values in the first-order Sobolev space $\mathbb{W}^{1}_{p}(\mathbb{R}^{d};\mathbb{R}^{d})$ provided $p>d$.

Article information

Source
Ann. Appl. Probab., Volume 26, Number 5 (2016), 2697-2732.

Dates
Received: June 2014
Revised: August 2015
First available in Project Euclid: 19 October 2016

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1476884301

Digital Object Identifier
doi:10.1214/15-AAP1159

Mathematical Reviews number (MathSciNet)
MR3563191

Zentralblatt MATH identifier
1353.60056

Subjects
Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60J60: Diffusion processes [See also 58J65]

Keywords
Weak differentiability Malliavin differentiability stability Krylov’s estimate Zvonkin’s transformation

Citation

Zhang, Xicheng. Stochastic differential equations with Sobolev diffusion and singular drift and applications. Ann. Appl. Probab. 26 (2016), no. 5, 2697--2732. doi:10.1214/15-AAP1159. https://projecteuclid.org/euclid.aoap/1476884301


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References

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