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Steady States of Fokker–Planck Equations: I. Existence

  • Wen Huang
  • Min Ji
  • Zhenxin Liu
  • Yingfei Yi
Article

Abstract

This is the first paper in a series concerning the study of steady states of a Fokker–Planck equation in a general domain in \(\mathbb {R}^n\) with \(L^{p}_{loc}\) drift term and \(W^{1,p}_{loc}\) diffusion term for any \(p>n\). In this paper, by using the level set method especially the integral identity which we introduced in Huang et al. (Ann Probab, 2015), we obtain several new existence results of steady states, including stationary solutions and measures, of the Fokker–Planck equation with non-degenerate diffusion under Lyapunov-like conditions. As applications of these results, we give some examples on the noise stabilization of an unstable equilibrium and the existence and uniqueness of steady states subject to boundary degeneracy of diffusion in a bounded domain.

Keywords

Fokker–Planck equation Existence Stationary solution Stationary measure Integral identity Level set method 

Mathematics Subject Classification

35Q84 60J60 37B25 60H10 

Notes

Acknowledgments

The first author was partially supported by NSFC Grant 11225105. The second author was partially supported by NSFC Innovation Grant 10421101. The third author was partially supported by NSFC Grant 11271151, SRF for ROCS, SEM, Chunmiao Fund and the 985 Program from Jilin University. The fourth author was partially supported by NSF Grants DMS0708331 and DMS1109201, a NSERC discovery grant, a faculty development grant from University of Alberta, and a Scholarship from Jilin University.

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Wu Wen-Tsun Key Laboratory of MathematicsUSTC, Chinese Academy of SciencesHefeiPeople’s Republic of China
  2. 2.Academy of Mathematics and System SciencesChinese Academy of SciencesBeijingPeople’s Republic of China
  3. 3.Department of Mathematical & Statistical SciencesUniversity of AlbertaEdmontonCanada
  4. 4.School of MathematicsJilin UniversityChangchunPeople’s Republic of China
  5. 5.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA

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