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Stochastic differential equations with coefficients in Sobolev spaces

文献类型：期刊论文

作者Fang, Shizan1; Luo, Dejun2,3; Thalmaier, Anton2
刊名JOURNAL OF FUNCTIONAL ANALYSIS
出版日期2010-09-01
卷号259期号:5页码:1129-1168
关键词Stochastic flows Sobolev space coefficients Density Density estimate Pathwise uniqueness Gaussian measure Ornstein-Uhlenbeck semigroup
ISSN号0022-1236
DOI10.1016/j.jfa.2010.02.014
英文摘要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.
WOS研究方向Mathematics
语种英语
WOS记录号WOS:000278700900003
出版者ACADEMIC PRESS INC ELSEVIER SCIENCE
源URL[http://ir.amss.ac.cn/handle/2S8OKBNM/9651]  
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.

入库方式： OAI收割

来源：数学与系统科学研究院

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