Stochastic stability of measures in gradient systems | |
Huang, Wen1,2; Ji, Min3; Liu, Zhenxin4; Yi, Yingfei5,6,7 | |
2016 | |
Source Publication | PHYSICA D-NONLINEAR PHENOMENA |
ISSN | 0167-2789 |
Volume | 314Pages:9-17 |
Abstract | Stochastic stability of a compact invariant set of a finite dimensional, dissipative system is studied in our recent work "Concentration and limit behaviors of stationary measures" (Huang et al., 2015) for general white noise perturbations. In particular, it is shown under some Lyapunov conditions that the global attractor of the systems is always stable under general noise perturbations and any strong local attractor in it can be stabilized by a particular family of noise perturbations. Nevertheless, not much is known about the stochastic stability of an invariant measure in such a system. In this paper, we will study the issue of stochastic stability of invariant measures with respect to a finite dimensional, dissipative gradient system with potential function f. As we will show, a special property of such a system is that it is the set of equilibria which is stable under general noise perturbations and the set S-f of global minimal points off which is stable under additive noise perturbations. For stochastic stability of invariant measures in such a system, we will characterize two cases Off, one corresponding to the case of finite Si. and the other one corresponding to the case when S-f is of positive Lebesgue measure, such that either some combined Dirac measures or the normalized Lebesgue measure on S-f is stable under additive noise perturbations. However, we will show by constructing an example that such measure stability can fail even in the simplest situation, i.e., in 1-dimension there exists a potential function f such that S-f consists of merely two points but no invariant measure of the corresponding gradient system is stable under additive noise perturbations. Crucial roles played by multiplicative and additive noise perturbations to the measure stability of a gradient system will also be discussed. In particular, the nature of instabilities of the normalized Lebesgue measure on S-f under multiplicative noise perturbations will be exhibited by an example. (C) 2015 Elsevier B.V. All rights reserved. |
Keyword | Fokker-Planck equation Gradient systems Gibbs measure Limit measure Stochastic stability White noise perturbation |
DOI | 10.1016/j.physd.2015.09.014 |
Language | 英语 |
Funding Project | NSFC[11225105] ; NSFC[11371339] ; NSFC[10421101] ; NSFC[11571344] ; NSFC[11271151] ; NSFC[11522104] ; Dalian University of Technology ; NSF[DMS1109201] ; NSERC[1257749] ; University of Alberta ; Jilin University |
WOS Research Area | Mathematics ; Physics |
WOS Subject | Mathematics, Applied ; Physics, Multidisciplinary ; Physics, Mathematical |
WOS ID | WOS:000366342000002 |
Publisher | ELSEVIER SCIENCE BV |
Citation statistics | |
Document Type | 期刊论文 |
Identifier | http://ir.amss.ac.cn/handle/2S8OKBNM/21464 |
Collection | 数学所 |
Affiliation | 1.Sichuan Univ, Sch Math, Chengdu 610064, Sichuan, Peoples R China 2.Univ Sci & Technol China, Sch Math Sci, Hefei 230026, Anhui, Peoples R China 3.Chinese Acad Sci, Acad Math & Syst Sci, Beijing 100080, Peoples R China 4.Dalian Univ Technol, Sch Math Sci, Dalian 116024, Peoples R China 5.Univ Alberta, Dept Math & Stat Sci, Edmonton, AB T6G 2G1, Canada 6.Jilin Univ, Sch Math, Changchun 130012, Peoples R China 7.Georgia Inst Technol, Sch Math, Atlanta, GA 30332 USA |
Recommended Citation GB/T 7714 | Huang, Wen,Ji, Min,Liu, Zhenxin,et al. Stochastic stability of measures in gradient systems[J]. PHYSICA D-NONLINEAR PHENOMENA,2016,314:9-17. |
APA | Huang, Wen,Ji, Min,Liu, Zhenxin,&Yi, Yingfei.(2016).Stochastic stability of measures in gradient systems.PHYSICA D-NONLINEAR PHENOMENA,314,9-17. |
MLA | Huang, Wen,et al."Stochastic stability of measures in gradient systems".PHYSICA D-NONLINEAR PHENOMENA 314(2016):9-17. |
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