SOBOLEV DIFFERENTIABLE STOCHASTIC FLOWS FOR SDES WITH SINGULAR COEFFICIENTS: APPLICATIONS TO THE TRANSPORT EQUATION

• 成果类型: Article
• 署名作者: Mohammed, Salah-Eldin A.; Nilssen, Torstein K.; Proske, Frank N.
• 署名单位: Southern Illinois University System; Southern Illinois University; University of Oslo
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/14-AOP909
• 发表日期: 2015
• 页码: 1535-1576
• 关键词: stable manifold theorem; uniqueness; systems; memory

摘要

In this paper, we establish the existence of a stochastic flow of Sobolev diffeomorphisms R-d (sic) x bar right arrow phi(s,t) (x) is an element of R-d, s,t is an element of R for a stochastic differential equation (SDE) of the form dX(t) = b(t, X-t)dt + dB(t), s,t is an element of R, X-s = x is an element of R-d. The above SDE is driven by a bounded measurable drift coefficient b: R x R-d -> R-d and a d-dimensional Brownian motion B. More specifically, we show that the stochastic flow phi(s,t) (center dot) of the SDE lives in the space L-2(Omega; W-1,W-P (R-d, w)) for ail s, t and all p is an element of (1, infinity), where W-1,W-p (R-d, w) denotes a weighted Sobolev space with weight w possessing a pth moment with respect to Lebesgue measure on 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 {d(t)u(t, x) + (b(t, x) center dot Du(t, x)) dt + Sigma(d)(i=1) e(i) center dot Du(t, x) circle dB(t)(i) = 0, u(0, x) = u(0)(x), where b is bounded and measurable, up is C-b(1), and {e(i)}(i=1)(d) a basis for R-d. It is well known that the deterministic counterpart of the above equation does not in general have a solution.