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Concentration and limit behaviors of stationary measures
Huang, Wen1; Ji, Min2,3,4; Liu, Zhenxin5; Yi, Yingfei6,7
2018-04-15
Source PublicationPHYSICA D-NONLINEAR PHENOMENA
ISSN0167-2789
Volume369Pages:1-17
AbstractIn this paper, we study limit behaviors of stationary measures of the Fokker-Planck equations associated with a system of ordinary differential equations perturbed by a class of multiplicative noise including additive white noise case. As the noises are vanishing, various results on the invariance and concentration of the limit measures are obtained. In particular, we show that if the noise perturbed systems admit a uniform Lyapunov function, then the stationary measures form a relatively sequentially compact set whose weak*-limits are invariant measures of the unperturbed system concentrated on its global attractor. In the case that the global attractor contains a strong local attractor, we further show that there exists a family of admissible multiplicative noises with respect to which all limit measures are actually concentrated on the local attractor; and on the contrary, in the presence of a strong local repeller in the global attractor, there exists a family of admissible multiplicative noises with respect to which no limit measure can be concentrated on the local repeller. Moreover, we show that if there is a strongly repelling equilibrium in the global attractor, then limit measures with respect to typical families of multiplicative noises are always concentrated away from the equilibrium. As applications of these results, an example of stochastic Hopf bifurcation and an example with non-decomposable omega-limit sets are provided. Our study is closely related to the problem of noise stability of compact invariant sets and invariant measures of the unperturbed system. (C) 2017 Elsevier B.V. All rights reserved.
KeywordFokker-Planck equation Stationary measure Limit measure Concentration Stochastic stability White noise perturbation
DOI10.1016/j.physd.2017.12.009
Language英语
Funding ProjectNSFC[11371339] ; NSFC[11431012] ; NSFC[11731003] ; NSFC[11571344] ; NSFC[11271151] ; NSFC[11522104] ; Dalian University of Technology ; NSF[DMS1109201] ; NSERC[1257749] ; University of Alberta ; Jilin University
WOS Research AreaMathematics ; Physics
WOS SubjectMathematics, Applied ; Physics, Multidisciplinary ; Physics, Mathematical
WOS IDWOS:000428831100001
PublisherELSEVIER SCIENCE BV
Citation statistics
Document Type期刊论文
Identifierhttp://ir.amss.ac.cn/handle/2S8OKBNM/30134
Collection数学所
Affiliation1.Univ Sci & Technol China, Sch Math Sci, Hefei 230026, Anhui, Peoples R China
2.Chinese Acad Sci, Acad Math & Syst Sci, Beijing 100080, Peoples R China
3.Chinese Acad Sci, Hua Loo Keng Key Lab Math, Beijing 100080, Peoples R China
4.Univ Chinese Acad Sci, Beijing 100080, Peoples R China
5.Dalian Univ Technol, Sch Math Sci, Dalian 116024, Peoples R China
6.Univ Alberta, Dept Math & Stat Sci, Edmonton, AB T6G 2G1, Canada
7.Jilin Univ, Sch Math, Changchun 130012, Jilin, Peoples R China
Recommended Citation
GB/T 7714
Huang, Wen,Ji, Min,Liu, Zhenxin,et al. Concentration and limit behaviors of stationary measures[J]. PHYSICA D-NONLINEAR PHENOMENA,2018,369:1-17.
APA Huang, Wen,Ji, Min,Liu, Zhenxin,&Yi, Yingfei.(2018).Concentration and limit behaviors of stationary measures.PHYSICA D-NONLINEAR PHENOMENA,369,1-17.
MLA Huang, Wen,et al."Concentration and limit behaviors of stationary measures".PHYSICA D-NONLINEAR PHENOMENA 369(2018):1-17.
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