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Exponential Ergodicity for SDEs Driven by α-Stable Processes with Markovian Switching in Wasserstein Distances

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Abstract

In this paper, we consider the ergodicity for stochastic differential equations driven by symmetric α-stable processes with Markovian switching in Wasserstein distances. Some sufficient conditions for the exponential ergodicity are presented by using the theory of M-matrix, coupling method and Lyapunov function method. As applications, the Ornstein-Uhlenbeck type process and some other processes driven by symmetric α-stable processes with Markovian switching are presented to illustrate our results. In addition, under some conditions, an explicit expression of the invariant measure for Ornstein-Uhlenbeck process is given.

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Acknowledgements

The authors are grateful to Professors Zhen-Qing Chen, Pengfei Yang and the anonymous reviewers for their valuable comments and suggestions which led to improvements in this manuscript. The research of J. Tong was partially supported by the National Natural Science Foundation of China (Nos. 11401093 and 11571071). The research of Z. Zhang is supported by the Humanities and Social Sciences Fund of Ministry of Education of China (No. 17YJA910004).

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Authors and Affiliations

  1. Department of Applied Mathematics, Donghua University, Shanghai, 201620, China

    Jinying Tong, Xinghu Jin & Zhenzhong Zhang

  2. Department of Mathematics, University of Macau, Macau, 999078, China

    Xinghu Jin

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  1. Jinying Tong
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  2. Xinghu Jin
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  3. Zhenzhong Zhang
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Correspondence to Zhenzhong Zhang.

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Tong, J., Jin, X. & Zhang, Z. Exponential Ergodicity for SDEs Driven by α-Stable Processes with Markovian Switching in Wasserstein Distances. Potential Anal 49, 503–526 (2018). https://doi.org/10.1007/s11118-017-9665-3

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  • DOI: https://doi.org/10.1007/s11118-017-9665-3

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