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

Author

Listed:
  • Zhi Li

    (Yangtze University
    Donghua University)

  • Litan Yan

    (Donghua University)

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 $$C_0$$ 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.

Suggested Citation

  • Zhi Li & Litan Yan, 2019. "Ergodicity and Stationary Solution for Stochastic Neutral Retarded Partial Differential Equations Driven by Fractional Brownian Motion," Journal of Theoretical Probability, Springer, vol. 32(3), pages 1399-1419, September.
    • Handle: RePEc:spr:jotpro:v:32:y:2019:i:3:d:10.1007_s10959-018-0810-8
    DOI: 10.1007/s10959-018-0810-8
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    References listed on IDEAS

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    1. Boufoussi, Brahim & Hajji, Salah, 2012. "Neutral stochastic functional differential equations driven by a fractional Brownian motion in a Hilbert space," Statistics & Probability Letters, Elsevier, vol. 82(8), pages 1549-1558.
    2. Liu, Kai, 2008. "Stationary solutions of retarded Ornstein-Uhlenbeck processes in Hilbert spaces," Statistics & Probability Letters, Elsevier, vol. 78(13), pages 1775-1783, September.
    3. Kai Liu, 2018. "Sensitivity to Small Delays of Pathwise Stability for Stochastic Retarded Evolution Equations," Journal of Theoretical Probability, Springer, vol. 31(3), pages 1625-1646, September.
    4. B. Pasik-Duncan & T. E. Duncan & B. Maslowski, 2006. "Linear Stochastic Equations in a Hilbert Space with a Fractional Brownian Motion," International Series in Operations Research & Management Science, in: Houmin Yan & George Yin & Qing Zhang (ed.), Stochastic Processes, Optimization, and Control Theory: Applications in Financial Engineering, Queueing Networks, and Manufacturing Systems, chapter 0, pages 201-221, Springer.
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