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Ergodicity and stationary solution for stochastic neutral retarded partial differential equations driven by fractional Brownian motion. (English) Zbl 1447.60114
Summary: 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.
MSC:
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60G15 Gaussian processes
60H05 Stochastic integrals
60G22 Fractional processes, including fractional Brownian motion
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[1] Alòs, E.; Mazet, O.; Nualart, D., Stochastic calculus with respect to Gaussian processes, Ann. Probab., 29, 766-801, (2001) · Zbl 1015.60047
[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) · Zbl 1248.60069
[3] Caraballo, T.; Garrido-Atienza, MJ; 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) · Zbl 1218.60053
[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) · Zbl 1317.60077
[5] Duncan, T.; Maslowski, B.; Pasik-Duncan, B., Fractional Brownian motion and stochastic equations in Hilbert space, Stoch. Dyn., 2, 225-250, (2002) · Zbl 1040.60054
[6] Duncan, T.; Maslowski, B.; Pasik-Duncan, B.; Hillier, FS (ed.); Price, CC (ed.), Linear stochastic Equations in a Hilbert Space with a Fractional Brownian Motion, No. 94, 201-222, (2006), Berlin
[7] Dung, NT, 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) · Zbl 1304.60074
[8] Garrido-Atienza, MJ; Kloeden, PE; Neuenkirch, A., Discretization of stationary solutions of stochastic systems driven by fractional Brownian motion, Appl. Math. Optim., 60, 151-172, (2009) · Zbl 1180.93095
[9] Hale, J.K., Lunel, S.M.: Introduction to Functional Differential Equations. Springer, New York (1993) · Zbl 0787.34002
[10] Kolmanovskii, V.B., Myshkis, A.: Introduction to the Theory and Applications of Functional Differential Equations. Kluwer Academic Publishers, Norwell (1999) · Zbl 0917.34001
[11] Li, Z., Global attractiveness and quasi-invariant sets of impulsive neutral stochastic functional differential equations driven by fBm, Neurocomputing, 177, 620-627, (2016)
[12] Lions, J.L., Magenes, E.: Nonhomogeneous Boundary Value Problems and Applications, vol. I, II and III. Spring, Berlin (1992)
[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)
[14] Liu, K., Stationary solutions of retarded Ornstein-Uhlenbeck processes in Hilbert spaces, Statist. Probab. Lett., 78, 1775-1783, (2008) · Zbl 1154.60053
[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)
[16] Liu, K., A Criterion for stationary solutions of retarded linear equations with additive noise, Stoch. Anal. Appl., 29, 799-823, (2011) · Zbl 1237.60051
[17] Liu, K., Quadratic control problem of neutral Ornstein-Uhlenbeck processes with control delays, Discret. Cont. Dyn. Syst. B, 18, 1651-1661, (2013) · Zbl 1401.60123
[18] Liu, K., Sensitivity to small delays of pathwise stability for stochastic retarded evolution equations, J. Theor. Probab., (2018) · Zbl 1404.60091
[19] Liu, K.: Stationary solutions of neutral stochastic partial differential equations with delays in the highest-order derivatives (2017) arXiv:1707.07827v1
[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) · Zbl 1388.60099
[21] Mandelbrot, BB; Ness, J., Fractional Brownian motion, fractional noises and applications, SIAM Rev., 10, 422-437, (1968) · Zbl 0179.47801
[22] Mao, X.R.: Stochastic Differential Equations and Applications, 2nd edn. Wood-head Publishing, Oxford (2007) · Zbl 1138.60005
[23] Maslowski, B.; Pospís̆il, J., Ergodicity and parameter estimates for infinite-dimensional fractional Ornstein-Uhlenbeck process, Appl. Math. Optim, 57, 401-429, (2008) · Zbl 1176.35185
[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) · Zbl 1062.60060
[25] Mohammed, S.-E.A.: Stochastic Functional Differential Equations. Piyman, Boston (1984) · Zbl 0584.60066
[26] Rozanov, Y.A.: Statinary Random Processes. Holden-Day, San Francisco (1996)
[27] Salamon, D.: Control and Observation of Neutral Systems, vol. 91. Pitman Advanced Publishing Program, London (1984) · Zbl 0546.93041
[28] Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives. Gordon and Breach Science Publishers, Yvendon (1993) · Zbl 0818.26003
[29] Wu, J.H.: Theory and Applications of Partial Functional Differential Equations, vol. 119. Springer, New York (1996) · Zbl 0870.35116
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