Exponential behavior and upper noise excitation index of solutions to evolution equations with unbounded delay and tempered fractional Brownian motions

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.

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References

  1. 1.

    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).

  2. 2.

    B. Blümich, White noise nonlinear system analysis in nuclear magnetic resonance spectroscopy, Prog. Nucl. Magn. Reson. Spectrosc. 19 (4) (1987) 331–417.

    Article  Google Scholar 

  3. 3.

    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.

    MathSciNet  Article  Google Scholar 

  4. 4.

    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.

    MathSciNet  Article  Google Scholar 

  5. 5.

    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.

    MathSciNet  Article  Google Scholar 

  6. 6.

    T. Caraballo, M. A. Hammami, L. Mchiri, Practical exponential stability of impulsive stochastic functional differential equations, Syst. Control Lett. 109 (2017) 43–48.

    MathSciNet  Article  Google Scholar 

  7. 7.

    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.

    MathSciNet  Article  Google Scholar 

  8. 8.

    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.

    MathSciNet  Article  Google Scholar 

  9. 9.

    R. F. Curtain, P. L. Falb, Stochastic differential equations in Hilbert space, J. Differ. Equ. 10 (3) (1971) 412–430.

    MathSciNet  Article  Google Scholar 

  10. 10.

    G. Da Prato, J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and Its Applications, Cambridge Univ. Press, Cambridge, MA, 1992.

    Google Scholar 

  11. 11.

    A. G. Davenport, The spectrum of horizontal gustiness near the ground in high winds, Q. J. R. Meteorol. Soc. 87 (1961) 194–211.

    Article  Google Scholar 

  12. 12.

    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.

    MathSciNet  MATH  Google Scholar 

  13. 13.

    M. Ferrante, C. Rovira, Convergence of delay differential equations driven by fractional Brownian motion, J. Evol. Equ. 10 (4) (2010) 761–783.

    MathSciNet  Article  Google Scholar 

  14. 14.

    M. Foondun, M. Joseph, Remarks on non-linear noise excitability of some stochastic heat equations, Stoch. Process. Appl. 124 (10) (2014) 3429–3440.

    MathSciNet  Article  Google Scholar 

  15. 15.

    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.

    MathSciNet  Article  Google Scholar 

  16. 16.

    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.

    MathSciNet  Article  Google Scholar 

  17. 17.

    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.

    MathSciNet  Article  Google Scholar 

  18. 18.

    J. J. Jang, J. S. Guo, Analysis of maximum wind force for offshore structure design, J. Mar. Sci. Technol. 7 (1) (1999) 43–51.

    Google Scholar 

  19. 19.

    D. Khoshnevisan, K. Kim, Nonlinear noise excitation of intermittent stochastic PDEs and the topology of LCA groups, Ann. Probab. 43 (4) (2015) 1944–1991.

    MathSciNet  Article  Google Scholar 

  20. 20.

    D. Khoshnevisan, K. Kim, Non-linear noise excitation and intermittency under high disorder, Proc. Am. Math. Soc. 143 (9) (2015) 4073–4083.

    MathSciNet  Article  Google Scholar 

  21. 21.

    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.

    MathSciNet  Article  Google Scholar 

  22. 22.

    Y. S. Li, A. Kareem, ARMA systems in wind engineering, Probab. Eng. Mech. 5 (2) (1990) 49–59.

    Article  Google Scholar 

  23. 23.

    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.

    MathSciNet  Article  Google Scholar 

  24. 24.

    B. Lindner, J. Garcia-Ojalvo, A. Neiman, L. Schimansky-Geier, Effects of noise in excitable systems, Phys. Rep. 392 (6) (2004) 321–424.

    Article  Google Scholar 

  25. 25.

    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.

    MathSciNet  Article  Google Scholar 

  26. 26.

    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.

    MathSciNet  Article  Google Scholar 

  27. 27.

    M. M. Meerschaert, F. Sabzikar, Tempered fractional Brownian motion, Stat. Probab. Lett. 83 (10) (2013) 2269–2275.

    MathSciNet  Article  Google Scholar 

  28. 28.

    M. M. Meerschaert, F. Sabzikar, Stochastic integration for tempered fractional Brownian motion, Stoch. Process. Appl. 124 (7) (2014) 2363–2387.

    MathSciNet  Article  Google Scholar 

  29. 29.

    A. Neuenkirch, I. Nourdin, S. Tindel, Delay equations driven by rough paths, Electron. J. Probab. 13 (67) (2008) 2031–2068.

    MathSciNet  Article  Google Scholar 

  30. 30.

    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.

    MathSciNet  Article  Google Scholar 

  31. 31.

    D. J. Norton, Mobile offshore platform wind loads, in: Proc. 13th Offshore Techn. Conf., OTC 4123, 4 (1981) 77–88.

  32. 32.

    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.

    MathSciNet  Article  Google Scholar 

  33. 33.

    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.

    MathSciNet  Article  Google Scholar 

  34. 34.

    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.

    MathSciNet  Google Scholar 

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Correspondence to Tomás Caraballo.

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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.

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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. (2021). https://doi.org/10.1007/s00028-020-00656-0

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Keywords

  • Stochastic PDEs
  • Unbounded delay
  • Tempered fractional Brownian motion
  • Fractional powers of closed operators
  • Exponential decay in mean square