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

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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 < ϵ ⩽ ϵ0} 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.

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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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Authors and Affiliations

  1. Mathematics and Science College, Shanghai Normal University, Shanghai, 200234, China

    Lifeng Chen & Jifa Jiang

  2. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Zhao Dong

  3. Wu Wen-Tsun Key Laboratory of Mathematics, University of Science and Technology of China, Hefei, 230026, China

    Jianliang Zhai

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  1. Lifeng Chen
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  2. Zhao Dong
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  3. Jifa Jiang
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  4. Jianliang Zhai
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Correspondence to Jifa Jiang.

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Chen, L., Dong, Z., Jiang, J. et al. On limiting behavior of stationary measures for stochastic evolution systems with small noise intensity. Sci. China Math. 63, 1463–1504 (2020). https://doi.org/10.1007/s11425-018-9527-1

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  • DOI: https://doi.org/10.1007/s11425-018-9527-1

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