Abstract
In this paper, we investigate stochastic evolution equations with unbounded delay in fractional power spaces perturbed by a tempered fractional Brownian motion \(B_Q^{\sigma ,\lambda }(t)\) with \(-1/2<\sigma <0\) and \(\lambda >0\). We first introduce a technical lemma which is crucial in our stability analysis. Then, we prove the existence and uniqueness of mild solutions by using semigroup methods. The upper nonlinear noise excitation index of the energy solutions at any finite time t is also obtained. Finally, we consider the exponential asymptotic behavior of mild solutions in mean square.
Similar content being viewed by others
References
J. P. P. Beaupuits, A. Otárola, F. T. Rantakyrö, R. C. Rivera, S. J. E. Radford, L.-Å Nyman, Analysis of wind data gathered at Chajnantor, ALMA Memo 497 (2004).
B. Blümich, White noise nonlinear system analysis in nuclear magnetic resonance spectroscopy, Prog. Nucl. Magn. Reson. Spectrosc. 19 (4) (1987) 331–417.
B. Boufoussi, S. Hajji, Neutral stochastic functional differential equations driven by a fractional Brownian motion in a Hilbert space, Statist. Probab. Lett. 82 (8) (2012) 1549–1558.
T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuss, J. Valero, Non-autonomous and random attractors for delay random semilinear equations without uniqueness, Discret. Contin. Dyn. Syst. 21 (2008) 415–443.
T. Caraballo, M. J. Garrido-Atienza, T. Taniguchi, The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion, Nonlinear Anal. 74 (11) (2011) 3671–3684.
T. Caraballo, M. A. Hammami, L. Mchiri, Practical exponential stability of impulsive stochastic functional differential equations, Syst. Control Lett. 109 (2017) 43–48.
G. L. Chen, D. S. Li, L. Shi, O. van Gaans, S. V. Lunel, Stability results for stochastic delayed recurrent neural networks with discrete and distributed delays, J. Differ. Equ. 264 (6) (2018) 3864–3898.
Y. Chen, X. D. Wang, W. H. Deng, Tempered fractional Langevin-Brownian motion with inverse \(\beta \)-stable subordinator, J. Phys. A: Math. Theor. 51 (2018) 495001.
R. F. Curtain, P. L. Falb, Stochastic differential equations in Hilbert space, J. Differ. Equ. 10 (3) (1971) 412–430.
G. Da Prato, J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and Its Applications, Cambridge Univ. Press, Cambridge, MA, 1992.
A. G. Davenport, The spectrum of horizontal gustiness near the ground in high winds, Q. J. R. Meteorol. Soc. 87 (1961) 194–211.
M. Ferrante, C. Rovira, Stochastic delay differential equations driven by fractional Brownian motion with Hurst parameter \(H>\frac{1}{2}\), Bernoulli 12 (2006) 85–100.
M. Ferrante, C. Rovira, Convergence of delay differential equations driven by fractional Brownian motion, J. Evol. Equ. 10 (4) (2010) 761–783.
M. Foondun, M. Joseph, Remarks on non-linear noise excitability of some stochastic heat equations, Stoch. Process. Appl. 124 (10) (2014) 3429–3440.
M. J. Garrido-Atienza, K. N. Lu, B. Schmalfuss, Local pathwise solutions to stochastic evolution equations driven by fractional Brownian motions with Hurst parameters \(H\in (1/3, 1/2]\), Discrete Contin. Dyn. Syst. Ser. B. 20 (8) (2015) 2553–2581.
M. J. Garrido-Atienza, K. N. Lu, B. Schmalfuss, Random dynamical systems for stochastic evolution equations driven by multiplicative fractional Brownian noise with Hurst parameters \(H \in [1/3, 1/2]\), SIAM J. Appl. Dyn. Syst. 15 (1) (2016) 625–654.
P. T. Hong, C. T. Binh, A note on exponential stability of non-autonomous linear stochastic differential delay equations driven by a fractional Brownian motion with Hurst index \(>\frac{1}{2}\), Stat. Probab. Lett. 138 (2018) 127–136.
J. J. Jang, J. S. Guo, Analysis of maximum wind force for offshore structure design, J. Mar. Sci. Technol. 7 (1) (1999) 43–51.
D. Khoshnevisan, K. Kim, Nonlinear noise excitation of intermittent stochastic PDEs and the topology of LCA groups, Ann. Probab. 43 (4) (2015) 1944–1991.
D. Khoshnevisan, K. Kim, Non-linear noise excitation and intermittency under high disorder, Proc. Am. Math. Soc. 143 (9) (2015) 4073–4083.
E. H. Lakhel, A. Tlidi, Controllability of time-dependent neutral stochastic functional differential equations driven by a fractional Brownian motion, J. Nonlinear Sci. Appl. 11 (2018), 850–863.
Y. S. Li, A. Kareem, ARMA systems in wind engineering, Probab. Eng. Mech. 5 (2) (1990) 49–59.
Y. J. Li, Y. J. Wang, The existence and asymptotic behavior of solutions to fractional stochastic evolution equations with infinite delay, J. Differ. Equ. 266 (6) (2019) 3514–3558.
B. Lindner, J. Garcia-Ojalvo, A. Neiman, L. Schimansky-Geier, Effects of noise in excitable systems, Phys. Rep. 392 (6) (2004) 321–424.
L. F. Liu, T. Caraballo, Analysis of a Stochastic 2D-Navier-Stokes Model with Infinite Delay, J. Dyn. Differ. Equ. 31 (4) (2019), 2249–2274.
W. Liu, K. H. Tian, M. Foondun, On some properties of a class of fractional stochastic heat equations, J. Theoret. Probab. 30 (4) (2017) 1310–1333.
M. M. Meerschaert, F. Sabzikar, Tempered fractional Brownian motion, Stat. Probab. Lett. 83 (10) (2013) 2269–2275.
M. M. Meerschaert, F. Sabzikar, Stochastic integration for tempered fractional Brownian motion, Stoch. Process. Appl. 124 (7) (2014) 2363–2387.
A. Neuenkirch, I. Nourdin, S. Tindel, Delay equations driven by rough paths, Electron. J. Probab. 13 (67) (2008) 2031–2068.
I. Norros, E. Valkeila, J. Virtamo, An elementary approach to a Girsanov formula and other analytical results on fractional Brownian motions, Bernoulli 5 (4) (1999) 571–587.
D. J. Norton, Mobile offshore platform wind loads, in: Proc. 13th Offshore Techn. Conf., OTC 4123, 4 (1981) 77–88.
T. Taniguchi, K. Liu, A. Truman, Existence, uniqueness, and asymptotic behavior of mild solutions to stochastic functional differential equations in Hilbert spaces, J. Differ. Equ. 181 (1) (2002) 72–91.
X. H. Wang, K. N. Lu, B. X. Wang, Exponential stability of non-autonomous stochastic delay lattice systems with multiplicative noise, J. Dyn. Differ. Equ. 28 (2016) 1309–1335.
L. P. Xua, J. W. Luo, Viability for stochastic functional differential equations in Hilbert spaces driven by fractional Brownian motion, Appl. Math. Comput. 341 (2019) 93–110.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This work was supported by NSF of China (Grant No. 41875084), the Fundamental Research Funds for the Central Universities under Grant Nos. lzujbky-2018-ot03 and lzujbky-2018-it58. The research of T. Caraballo has been partially supported by Ministerio de Ciencia Innovación y Universidades (Spain), FEDER (European Community) under Grant PGC2018-096540-B-I00, and by Junta de Andalucía (Consejería de Economía y Conocimiento) and FEDER under Projects US-1254251 and P18-FR-4509.
Rights and permissions
About this article
Cite this article
Wang, Y., Liu, Y. & Caraballo, T. Exponential behavior and upper noise excitation index of solutions to evolution equations with unbounded delay and tempered fractional Brownian motions. J. Evol. Equ. 21, 1779–1807 (2021). https://doi.org/10.1007/s00028-020-00656-0
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00028-020-00656-0