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Steady States of Fokker-Planck Equations: II. Non-existence
Huang, Wen1,2; Ji, Min3; Liu, Zhenxin4,6; Yi, Yingfei5,6,7
AbstractThis is the second paper in a series concerning the study of steady states, including stationary solutions and measures, of a Fokker-Planck equation in a general domain in with drift term and diffusion term for any . In this paper, we obtain some non-existence results of stationary measures under conditions involving anti-Lyapunov type of functions associated with the stationary Fokker-Planck equation. When combined with the existence results showed in part I of the series (Huang et al. in J. Dyn Differ Equ) contained in the same volume, not only will these results yield necessary and sufficient conditions for the existence of stationary measures, but also they provide a useful tool for one to study noise perturbations of systems of ordinary differential equations, especially with respect to problems of stochastic bifurcations, as demonstrated in some examples contained in this paper. Our analysis is based on the level set method, in particular the integral identity, and measure estimates contained in our work (Huang et al. in Ann Probab 43:1712-1730, 2015).
KeywordFokker-Planck equation Non-existence Stationary solution Stationary measure Level set method
Funding ProjectNSFC[11225105] ; NSFC[11431012] ; NSFC[11271151] ; NSFC Innovation Grant[10421101] ; startup fund of Dalian University of Technology ; NSF[DMS0708331] ; NSF[DMS1109201] ; NSERC[1257749] ; University of Alberta ; Jilin University
WOS Research AreaMathematics
WOS SubjectMathematics, Applied ; Mathematics
WOS IDWOS:000366644300017
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Document Type期刊论文
Corresponding AuthorJi, Min
Affiliation1.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. Steady States of Fokker-Planck Equations: II. Non-existence[J]. JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS,2015,27(3-4):743-762.
APA Huang, Wen,Ji, Min,Liu, Zhenxin,&Yi, Yingfei.(2015).Steady States of Fokker-Planck Equations: II. Non-existence.JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS,27(3-4),743-762.
MLA Huang, Wen,et al."Steady States of Fokker-Planck Equations: II. Non-existence".JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS 27.3-4(2015):743-762.
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