Stochastic differential equations with coefficients in Sobolev spaces

doi:10.1016/j.jfa.2010.02.014

CSpace > 应用数学研究所

	Stochastic differential equations with coefficients in Sobolev spaces
	Fang, Shizan 1; Luo, Dejun2,3 ; Thalmaier, Anton 2
	2010-09-01
发表期刊	JOURNAL OF FUNCTIONAL ANALYSIS
ISSN	0022-1236
卷号	259 期号:5 页码:1129-1168
摘要	We consider the Ito stochastic differential equation dX(t) = Sigma(m)(j=1) A(j) dw(i)(j) + A0(X(i)) dt on R(d). The diffusion coefficients A1,..., A(m) are supposed to be in the Sobolev space W(loc)(i,p)(R(d)) with p > d, and to have linear growth. For the drift coefficient Ao, we distinguish two cases: (i) A(0) is a continuous vector field whose distributional divergence delta(A(0)) with respect to the Gaussian measure gamma(d) exists, (ii) A(0) has Sobolev regularity W(loc)(i,p) for some p' > I. Assume f(R)d exp[lambda(0)(vertical bar delta(A(0))vertical bar EniL (16(Ai)12 + vAi 12))1 dvd < +00 for some lambda(0) > 0. In case (i), if the pathwise uniqueness of solutions holds, then the push-forward (X(t))#gamma d admits a density with respect to gamma(d). In particular, if the coefficients are bounded Lipschitz continuous, then X(t) leaves the Lebesgue measure Leb(d) quasi-invariant. In case (ii), we develop a method used by G. Crippa and C. De Lellis for ODE and implemented by X. Zhang for SDE, to establish existence and uniqueness of stochastic flow of maps. (C) 2010 Elsevier Inc. All rights reserved.
关键词	Stochastic flows Sobolev space coefficients Density Density estimate Pathwise uniqueness Gaussian measure Ornstein-Uhlenbeck semigroup
DOI	10.1016/j.jfa.2010.02.014
语种	英语
WOS研究方向	Mathematics
WOS类目	Mathematics
WOS记录号	WOS:000278700900003
出版者	ACADEMIC PRESS INC ELSEVIER SCIENCE
引用统计
文献类型	期刊论文
条目标识符	http://ir.amss.ac.cn/handle/2S8OKBNM/9652
专题	应用数学研究所
通讯作者	Fang, Shizan
作者单位	1.Univ Bourgogne, IMB, Dijon, France 2.Univ Luxembourg, UR Math, L-1359 Luxembourg, Luxembourg 3.Chinese Acad Sci, Key Lab Random Complex Struct & Data Sci, Acad Math & Syst Sci, Beijing 100190, Peoples R China
推荐引用方式 GB/T 7714	Fang, Shizan,Luo, Dejun,Thalmaier, Anton. Stochastic differential equations with coefficients in Sobolev spaces[J]. JOURNAL OF FUNCTIONAL ANALYSIS,2010,259(5):1129-1168.
APA	Fang, Shizan,Luo, Dejun,&Thalmaier, Anton.(2010).Stochastic differential equations with coefficients in Sobolev spaces.JOURNAL OF FUNCTIONAL ANALYSIS,259(5),1129-1168.
MLA	Fang, Shizan,et al."Stochastic differential equations with coefficients in Sobolev spaces".JOURNAL OF FUNCTIONAL ANALYSIS 259.5(2010):1129-1168.