Global attracting sets and stability of neutral stochastic functional differential equations driven by Rosenblatt process

Abstract

We are concerned with a class of neutral stochastic partial differential equations driven by Rosenblatt process in a Hilbert space. By combining some stochastic analysis techniques, tools from semigroup theory, and stochastic integral inequalities, we identify the global attracting sets of this kind of equations. Especially, some sufficient conditions ensuring the exponent p-stability of mild solutions to the stochastic systems under investigation are obtained. Last, an example is given to illustrate the theory in the work.

Keywords

Global attracting sets / exponential p-th moment stability / Rosenblatt process

References

[1] Albin J M P. A note on Rosenblatt distributions. Statist. Probab. Lett, 1998, 40(1): 83–91

[2] Bardet J M, Tudor C A. A wavelet analysis of the Rosenblatt process: chaos expansion and estimation of the self-similarity parameter. Stochastic Process Appl, 2010, 120(12): 2331–2362

[3] 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

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

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

[6] 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

[7] Long S, Teng L, Xu D Y. Global attracting set and stability of stochastic neutral partial functional differential equations with impulses. Statist Probab Lett, 2012, 82: 1699–1709

[8] Maejima M, Tudor C A. Wiener integrals with respect to the Hermite process and a non central limit theorem. Stochastic Anal Appl, 2007, 25: 1043–1056

[9] Maejima M, Tudor C A. On the distribution of the Rosenblatt process. Statist Probab Lett, 2013, 83(6): 1490–1495

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

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

[12] Shen G J, Ren Y. Neutral stochastic partial differential equations with delay driven by Rosenblatt process in a Hilbert space. J Korean Statist Soc, 2015, 44: 123–133

[13] Shen G J, Yin X, Yan L T. Approximation of the Rosenblatt sheet. Mediterr J Math, 2016, 13: 2215–2227

[14] Shen G J, Yin X, Zhu D J. Weak convergence to Rosenblatt sheet. Front Math China, 2015, 10: 985–1004

[15] Shieh N R, Xiao Y M. Hausdorff and packing dimensions of the images of random fields. Bernoulli, 2010, 16(4): 926–952

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

[17] Taqqu M S. Weak convergence to fractional Brownian motion and to the Rosenblatt process. Probab Theory Related Fields, 1975, 31(4): 287–302

[18] Tudor C A. Analysis of the Rosenblatt process. ESAIM Probab Stat, 2008, 12: 230–257

[19] Tudor C A, Viens F G. Variations and estimators for self-similarity parameters via Malliavin calculus. Ann Probab, 2009, 37(6): 2093–2134

[20] Veillette M, Taqqu M S. Properties and numerical evaluation of the Rosenblatt distribution. Bernoulli, 2013, 19: 982–1005

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

[22] Xu D Y, Long S J. Attracting and quasi-invariant sets of no-autonomous neutral networks with delays. Neurocomputing, 2012, 77: 222–228

[23] Zhao Z H, Jian J G. Attracting and quasi-invariant sets for BAM neural networks of neutral-type with time-varying and infinite distributed delays. Neurocomputing, 2014, 140: 265–272

RIGHTS & PERMISSIONS

Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature