Ergodicity and Stationary Solution for Stochastic Neutral Retarded Partial Differential Equations Driven by Fractional Brownian Motion
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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.
KeywordsFractional Brownian motion Stochastic functional equation of neutral type Stationary solution Erodicity
Mathematics Subject Classification (2010)60H15 60G15 60H05
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.
- 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
- 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
- 12.Lions, J.L., Magenes, E.: Nonhomogeneous Boundary Value Problems and Applications, vol. I, II and III. Spring, Berlin (1992)Google Scholar
- 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
- 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
- 19.Liu, K.: Stationary solutions of neutral stochastic partial differential equations with delays in the highest-order derivatives (2017) arXiv:1707.07827v1
- 26.Rozanov, Y.A.: Statinary Random Processes. Holden-Day, San Francisco (1996)Google Scholar