Global attractiveness and exponential decay of neutral stochastic functional differential equations driven by fBm with Hurst parameter less than 1/2

Abstract

We are concerned with a class of neutral stochastic functional differential equations driven by fractional Brownian motion (fBm) in the Hilbert space. We obtain the global attracting sets of this kind of equations driven by fBm with Hurst parameter \hslash \in (0, 1/2): Especially, some suffcient conditions which ensure the exponential decay in the p-th moment of the mild solution of the considered equations are obtained. In the end, one example is given to illustrate the feasibility and effectiveness of results obtained.

Keywords

Global attracting sets / exponential p-th moment stability / fractional Brownian motion (fBm)

References

[1] Boudrahem S, Rougier P R. Relation between postural control assessment with eyes open and centre of pressure visual feed back effects in healthy individuals. Exp Brain Res, 2009, 195: 145–152

[2] Boufoussi B, Hajji S. Neutral stochastic functional differential equations driven by a fractional Brownian motion in a Hilbert space. Statist Probab Lett, 2012, 82(8): 1549–1558

[3] Boufoussi B, Hajji S. Transportation inequalities for neutral stochastic differential equations driven by fractional Brownian motion with Hurst parameter lesser than 1/2. Mediterr J Math, 2017, 14: 192

[4] 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, 2011, 74: 3671–3684

[5] Comte F, Renault E. Long memory continuous time models. J Econometrics, 1996, 73: 101–149

[6] de la Fuente I M, Perez-Samartin A L, Martinez L, Garcia M A, Vera-Lopez A. Long-range correlations in rabbit brain neural activity. Ann Biomed Eng, 2006, 34(2): 295–299

[7] Duncan T E, Maslowski B, Pasik-Duncan B. Fractional Brownian motion and stochastic equations in Hilbert spaces. Stoch Dyn, 2002, 2: 225–250

[8] Hale J K, Lunel S M. Introduction to Functional Differential Equations. New York: Springer-Verlag, 1993

[9] Kolmanovskii V B, Myshkis A. Introduction to the Theory and Applications of Functional Differential Equations. Kluwer Academic Publishers, 1999

[10] Lakhel El H. Controllability of neutral stochastic functional integro-differential equations driven by fractional Brownian motion. Stoch Anal Appl, 2016, 34(3): 427–440

[11] Lakhel El H. Controllability of neutral functional differential equations driven by fractional Brownian motion with innite delay. Nonlinear Dyn Syst Theory, 2017, 17(3): 291–302

[12] Lakhel El H, McKibben M A. Existence of solutions for fractional neutral functional differential equations driven by fBm with innite delay. Stochastics, 2018, 90(3): 313–329

[13] Li D S, Xu D Y. Attracting and quasi-invariant sets of stochastic neutral partial functional differential equations. Acta Math Sci Ser B Engl Ed, 2013, 33: 578–588

[14] Li Z. Global attractiveness and quasi-invariant sets of impulsive neutral stochastic functional differential equations driven by fBm. Neurocomputing, 2016, 177: 620–627

[15] Li Z, Luo J W. Transportation inequalities for stochastic delay evolution equations driven by fractional Brownian motion. Front Math China, 2015, 10(2): 303–321

[16] Liu K, Li Z. Global attracting set, exponential decay and stability in distribution of neutral SPDEs driven by additive α-stable processes. Discrete Contin Dyn Syst Ser B, 2016, 21: 3551–3573

[17] Mandelbrot B B, Van Ness J. Fractional Brownian motion, fractional noises and applications. SIAM Rev, 1968, 10: 422–437

[18] Mao X R. Stochastic Differential Equations and Applications. 2nd ed. Oxford: Woodhead Publishing, 2007

[19] Maslowski B, Nualart D. Evolution equations driven by a fractional Brownian motion. J Funct Anal, 2003, 202: 277–305

[20] Mohammed S-E A. Stochastic Functional Differential Equations. Boston: Pitman, 1984

[21] Nualart D. The Malliavin Calculus and Related Topics. 2nd ed. Berlin: Springer-Verlag, 2006

[22] Pazy A. Semigroup of Linear Operators and Applications to Partial Differential Equations. New York: Spring-Verlag, 1992

[23] Rypdal M, Rypdal K. Testing hypotheses about sun-climate complexity linking. Phys Rev Lett, 2010, 104: 128–151

[24] Salamon D. Control and Observation of Neutral Systems. Research Notes in Math, Vol 91. London: Pitman Advanced Publishing Program, 1984

[25] Simonsen I. Measuring anti-correlations in the nordic electricity spot market by wavelets. Phys A, 2003, 322: 597{606

[26] Wang L, Li D. Impulsive-integral inequalities for attracting and quasi-invariant sets of impulsive stochastic partial differential equations with innite delays. J Inequal Appl, 2013, 2013: 338

[27] Willinger W, Leland W, Taqqu M, Wilson D. On self-similar nature of ethernet trac. IEEE/ACM Trans Networking, 1994, 2: 1–15

[28] Wu J H. Theory and Applications of Partial Functional Differential Equations. Appl Math Sci, Vol 119. New York: Springer-Verlag, 1996

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