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Stochastic Liénard equations with state-dependent switching

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Abstract

This work focuses on stochastic Liénard equations with state-dependent switching. First, the existence and uniqueness of a strong solution are obtained by successive construction method. Next, strong Feller property is proved by introducing certain auxiliary processes and using the Radon-Nikodym derivatives and truncation arguments. Based on these results, positive Harris recurrence and exponential ergodicity are obtained under the Foster-Lyapunov drift conditions. Finally, examples using van der Pol equations are presented for illustrations, and the corresponding Foster-Lyapunov functions for the examples are constructed explicitly.

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Authors and Affiliations

  1. School of Mathematics, Beijing Institute of Technology, Beijing, 100081, China

    Fu-bao Xi

  2. Department of Mathematics, Wayne State University, Detroit, MI, 48202, USA

    G. Yin

Authors
  1. Fu-bao Xi
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  2. G. Yin
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Correspondence to Fu-bao Xi.

Additional information

Supported by the National Natural Science Foundation of China (No. 11171024) and the National Science Foundation, United States (No. DMS-0907753).

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Xi, Fb., Yin, G. Stochastic Liénard equations with state-dependent switching. Acta Math. Appl. Sin. Engl. Ser. 31, 893–908 (2015). https://doi.org/10.1007/s10255-015-0538-5

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  • DOI: https://doi.org/10.1007/s10255-015-0538-5

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