The Annals of Applied Probability

Stochastic differential equations with Sobolev diffusion and singular drift and applications

Xicheng Zhang

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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

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

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

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Zentralblatt MATH identifier

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

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


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.

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