zbMATH — the first resource for mathematics

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.
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60G15 Gaussian processes
60H05 Stochastic integrals
60G22 Fractional processes, including fractional Brownian motion
Full Text: DOI
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.