## The Annals of Applied Probability

### Stochastic differential equations with Sobolev diffusion and singular drift and applications

Xicheng Zhang

#### 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
Revised: August 2015
First available in Project Euclid: 19 October 2016

https://projecteuclid.org/euclid.aoap/1476884301

Digital Object Identifier
doi:10.1214/15-AAP1159

Mathematical Reviews number (MathSciNet)
MR3563191

Zentralblatt MATH identifier
1353.60056

#### 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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