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Ergodicity and Stationary Solution for Stochastic Neutral Retarded Partial Differential Equations Driven by Fractional Brownian Motion

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Abstract

In this paper, we discuss a class of neutral retarded stochastic functional differential equations driven by a fractional Brownian motion on Hilbert spaces. We develop a \(C_0\)-semigroup theory of the driving deterministic neutral system and formulate the neutral time delay equation under consideration as an infinite-dimensional stochastic system without time lag and neutral item. Consequently, a criterion is presented to identify a strictly stationary solution for the systems considered. In particular, the ergodicity of the strictly stationary solution is studied. Subsequently, the ergodicity behavior of non-stationary solution for the systems considered is also investigated. We present an example which can be explicitly determined to illustrate our theory in the work.

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Acknowledgements

The first author was supported by the Natural Science Foundation of Hubei Province (No. 2016CFB479) and China postdoctoral fund (No. 2017M610216). The second author was supported by the NNSF of China (No. 11571071). The authors would like to thank the Editor for carefully handling our paper and the anonymous referees for their valuable comments and suggestions for improving the quality of this work.

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Authors and Affiliations

  1. School of Mathematics and Information Sciences, Yangtze University, Jingzhou, 434023, Hubei, People’s Republic of China

    Zhi Li

  2. College of Information Science and Technology, Donghua University, Songjiang, 201620, Shanghai, People’s Republic of China

    Zhi Li & Litan Yan

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  1. Zhi Li
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  2. Litan Yan
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Correspondence to Litan Yan.

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Li, Z., Yan, L. Ergodicity and Stationary Solution for Stochastic Neutral Retarded Partial Differential Equations Driven by Fractional Brownian Motion. J Theor Probab 32, 1399–1419 (2019). https://doi.org/10.1007/s10959-018-0810-8

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  • DOI: https://doi.org/10.1007/s10959-018-0810-8

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