Advertisement

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

Article
  • 88 Downloads

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.

Keywords

Fractional Brownian motion Stochastic functional equation of neutral type Stationary solution Erodicity 

Mathematics Subject Classification (2010)

60H15 60G15 60H05 

Notes

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.

References

  1. 1.
    Alòs, E., Mazet, O., Nualart, D.: Stochastic calculus with respect to Gaussian processes. Ann. Probab. 29, 766–801 (2001)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Boufoussi, B., Hajji, S.: Neutral stochastic functional differential equations driven by a fractional Brownian motion in a Hilbert space. Stat. Probab. Lett. 82, 1549–1558 (2012)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Caraballo, T., Garrido-Atienza, M.J., Taniguchi, T.: The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion. Nonlinear Anal. 74, 3671–3684 (2011)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Da Prato, G., Zabczyk, J. (eds.): Stochastic Equations in Infinite Dimensions. In: Encyclopedia of Mathematics and its Application, 2nd edn. Cambridge University Press, Cambridge (2014)Google Scholar
  5. 5.
    Duncan, T., Maslowski, B., Pasik-Duncan, B.: Fractional Brownian motion and stochastic equations in Hilbert space. Stoch. Dyn. 2(2), 225–250 (2002)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Duncan, T., Maslowski, B., Pasik-Duncan, B.: Linear stochastic Equations in a Hilbert Space with a Fractional Brownian Motion. In: Hillier, F.S., Price, C.C. (eds.) International Series in Operations and Management Science, vol. 94, pp. 201–222. Springer, Berlin (2006)Google Scholar
  7. 7.
    Dung, N.T.: Neutral stochastic differential equations driven by a fractional Brownian motion with impulsive effects and varying-time delays. J. Korean Stat. Soc. 43, 599–608 (2014)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Garrido-Atienza, M.J., Kloeden, P.E., Neuenkirch, A.: Discretization of stationary solutions of stochastic systems driven by fractional Brownian motion. Appl. Math. Optim. 60, 151–172 (2009)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Hale, J.K., Lunel, S.M.: Introduction to Functional Differential Equations. Springer, New York (1993)CrossRefMATHGoogle Scholar
  10. 10.
    Kolmanovskii, V.B., Myshkis, A.: Introduction to the Theory and Applications of Functional Differential Equations. Kluwer Academic Publishers, Norwell (1999)CrossRefMATHGoogle Scholar
  11. 11.
    Li, Z.: Global attractiveness and quasi-invariant sets of impulsive neutral stochastic functional differential equations driven by fBm. Neurocomputing 177, 620–627 (2016)CrossRefGoogle Scholar
  12. 12.
    Lions, J.L., Magenes, E.: Nonhomogeneous Boundary Value Problems and Applications, vol. I, II and III. Spring, Berlin (1992)Google Scholar
  13. 13.
    Liu, K.: Stability of Infinite Dimensional Stochastic Diferential Equations with Applications. In: Monographs and Surveys in Pure and Applied Mathematics, vol. 135. Chapman and Hall, London (2006)Google Scholar
  14. 14.
    Liu, K.: Stationary solutions of retarded Ornstein–Uhlenbeck processes in Hilbert spaces. Statist. Probab. Lett. 78, 1775–1783 (2008)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Liu, K.: Finite pole assignment of linear neutral systems in infinite dimensions, In: Jiang, Y., Chen, X.G. (eds.) Proceedings of the Second International Conference on Modeling and Simulation (ICMS2009), pp. 1–11. Manchester (2009)Google Scholar
  16. 16.
    Liu, K.: A Criterion for stationary solutions of retarded linear equations with additive noise. Stoch. Anal. Appl. 29, 799–823 (2011)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Liu, K.: Quadratic control problem of neutral Ornstein–Uhlenbeck processes with control delays. Discret. Cont. Dyn. Syst. B 18, 1651–1661 (2013)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Liu, K.: Sensitivity to small delays of pathwise stability for stochastic retarded evolution equations. J. Theor. Probab. (2018).  https://doi.org/10.1007/s10959-017-0750-8 Google Scholar
  19. 19.
    Liu, K.: Stationary solutions of neutral stochastic partial differential equations with delays in the highest-order derivatives (2017) arXiv:1707.07827v1
  20. 20.
    Liu, K.: Almost sure exponential stability sensitive to small time delay of stochastic neutral functional differential equations. Appl. Math. Lett. 77, 57–63 (2018)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Mandelbrot, B.B., Van Ness, J.: Fractional Brownian motion, fractional noises and applications. SIAM Rev. 10, 422–437 (1968)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Mao, X.R.: Stochastic Differential Equations and Applications, 2nd edn. Wood-head Publishing, Oxford (2007)MATHGoogle Scholar
  23. 23.
    Maslowski, B., Pospís̆il, J.: Ergodicity and parameter estimates for infinite-dimensional fractional Ornstein–Uhlenbeck process. Appl. Math. Optim 57, 401–429 (2008)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Maslowski, B., Schmalfuss, B.: Random dynamical systems and stationary solutions of differential equations driven by the fractional Brownian motion. Stoch. Anal. Appl. 22, 1577–1607 (2004)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Mohammed, S.-E.A.: Stochastic Functional Differential Equations. Piyman, Boston (1984)MATHGoogle Scholar
  26. 26.
    Rozanov, Y.A.: Statinary Random Processes. Holden-Day, San Francisco (1996)Google Scholar
  27. 27.
    Salamon, D.: Control and Observation of Neutral Systems, vol. 91. Pitman Advanced Publishing Program, London (1984)MATHGoogle Scholar
  28. 28.
    Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives. Gordon and Breach Science Publishers, Yvendon (1993)MATHGoogle Scholar
  29. 29.
    Wu, J.H.: Theory and Applications of Partial Functional Differential Equations, vol. 119. Springer, New York (1996)MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematics and Information SciencesYangtze UniversityJingzhouPeople’s Republic of China
  2. 2.College of Information Science and TechnologyDonghua UniversitySongjiangPeople’s Republic of China

Personalised recommendations