STOCHASTIC DIFFERENTIAL EQUATIONS WITH SOBOLEV DIFFUSION AND SINGULAR DRIFT AND APPLICATIONS

• 成果类型: Article
• 署名作者: Zhang, Xicheng
• 署名单位: Wuhan University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/15-AAP1159
• 发表日期: 2016
• 页码: 2697-2732
• 关键词: sdes; FLOWS

摘要

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 R-d with initial values in the first-order Sobolev space w(P)(1) (R-d; R-d) provided p > d.