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On limiting behavior of stationary measures for stochastic evolution systems with small noise intensity

  • Lifeng Chen
  • Zhao Dong
  • Jifa JiangEmail author
  • Jianliang Zhai
Articles
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Abstract

The limiting behavior of stochastic evolution processes with small noise intensity ϵ is investigated in distribution-based approaches. Let μϵ be a stationary measure for stochastic process Xϵ with small ϵ and X0 be a semiflow on a Polish space. Assume that {μϵ: 0 < ϵϵ0} is tight. Then all their limits in the weak sense are X0-invariant and their supports are contained in the Birkhoff center of X0. Applications are made to various stochastic evolution systems, including stochastic ordinary differential equations, stochastic partial differential equations, and stochastic functional differential equations driven by Brownian motion or Lévy processes.

Keywords

stationary measure Lyapunov function limit measure support Birkhoff center stochastic evolution system 

MSC(2010)

60B10 60G10 60H30 60H10 60H15 34K50 37L40 37A25 

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Notes

Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11771295, 11271356, 11371041, 11431014 and 11401557), Key Laboratory of Random Complex Structures and Data Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, and the Fundamental Research Funds for the Central Universities (Grant No. WK0010000048).

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Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Lifeng Chen
    • 1
  • Zhao Dong
    • 2
  • Jifa Jiang
    • 1
    Email author
  • Jianliang Zhai
    • 3
  1. 1.Mathematics and Science CollegeShanghai Normal UniversityShanghaiChina
  2. 2.Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingChina
  3. 3.Wu Wen-Tsun Key Laboratory of MathematicsUniversity of Science and Technology of ChinaHefeiChina

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