\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents
Early Access

Early Access articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Early Access publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Early Access articles via the “Early Access” tab for the selected journal.

Asymptotic behaviour of time fractional stochastic delay evolution equations with tempered fractional noise

  • *Corresponding author

    *Corresponding author

This work was supported by the National Natural Science Foundation of China under grant 41875084

Abstract Full Text(HTML) Related Papers Cited by
  • This paper is concerned with stochastic delay evolution equations driven by tempered fractional Brownian motion (tfBm) $ B_Q^{\sigma, \lambda}(t) $ with time fractional operator of order $ \alpha\in (1/2+\sigma, 1) $, where $ \sigma\in (-1/2, 0) $ and $ \lambda>0 $. First, we establish the global existence and uniqueness of mild solutions by using the new established estimation of stochastic integrals with respect to tfBm. Moreover, based on the relations between the stochastic integrals with respect to tfBm and fBm, we show the continuity of mild solutions for stochastic delay evolution equations when tempered fractional noise is reduced to fractional noise. Finally, we analyze the stability with general decay rate (including exponential, polynomial and logarithmic stability) of mild solutions for stochastic delay evolution equations with tfBm and time tempered fractional operator.

    Mathematics Subject Classification: Primary: 34K50, 60G22; Secondary: 34K20, 34K37.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] D. Baleanu, K. Diethelm, E. Scalas and J. J. Trujillo, Fractional calculus: Models and numerical methods, World Scientific, Boston, 2012. doi: 10.1142/9789814355216.
    [2] O. E. Barndorff-Nielsen, Processes of normal inverse Gaussian type, Finance Stoch., 2 (1998), 41-68.  doi: 10.1007/s007800050032.
    [3] A. Benchaabane and R. Sakthivel, Sobolev-type fractional stochastic differential equations with non-Lipschitz coefficients, J. Comput. Appl. Math., 312 (2017), 65-73.  doi: 10.1016/j.cam.2015.12.020.
    [4] B. Boufoussi and S. Hajji, Neutral stochastic functional differential equations driven by a fractional Brownian motion in a Hilbert space, Statist. Probab. Lett., 82 (2012), 1549-1558.  doi: 10.1016/j.spl.2012.04.013.
    [5] T. CaraballoM. J. Garrido-AtienzaB. Schmalfuß and J. Valero, Non-autonomous and random attractors for delay random semilinear equations without uniqueness, Discrete Contin. Dyn. Syst., 21 (2008), 415-443.  doi: 10.3934/dcds.2008.21.415.
    [6] A. G. Davenport, The spectrum of horizontal gustiness near the ground in high winds, Q. J. R. Meteorol. Soc., 87 (1961), 194-211. 
    [7] B. De AndradeA. N. CarvalhoP. M. Carvalho-Neto and P. Marín-Rubio, Semilinear fractional differential equations: Global solutions, critical nonlinearities and comparison results, Topol. Methods Nonlinear Anal., 45 (2015), 439-467.  doi: 10.12775/TMNA.2015.022.
    [8] P. M. De Carvalho-Neto and G. Planas, Mild solutions to the time fractional Navier-Stokes equations in $\mathbb{R}^N$, J. Differential Equations, 259 (2015), 2948-2980.  doi: 10.1016/j.jde.2015.04.008.
    [9] R. FriedrichF. JenkoA. Baule and S. Eule, Anomalous diffusion of inertial, weakly damped particles, Phys. Rev. Lett., 96 (2006), 230601.  doi: 10.1103/PhysRevLett.96.230601.
    [10] L. GiraitisP. Kokoszka and R. Leipus, Stationary ARCH models: Dependence structure and central limit theorem, Econometric Theory, 16 (2000), 3-22.  doi: 10.1017/S0266466600161018.
    [11] Y. GuoM. ChenX.-B. Shu and F. Xu, The existence and Hyers-Ulam stability of solution for almost periodical fractional stochastic differential equation with fBm, Stoch. Anal. Appl., 39 (2021), 643-666.  doi: 10.1080/07362994.2020.1824677.
    [12] P. T. Huong, P. E. Kloeden and D. T. Son, Well-posedness and regularity for solutions of Caputo stochastic fractional differential equations in $L^p$ spaces, Stoch Anal. Appl., (2021), 1-15. doi: 10.1080/07362994.2021.1988856.
    [13] J.-J. Jang and G. Jyh-Shinn, Analysis of maximum wind force for offshore structure design, J. Marine Sci. Tech., 7 (1999), 43-51.  doi: 10.51400/2709-6998.2511.
    [14] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier. Science. B.V., Amsterdam, 2006.
    [15] Y. Li and Y. Wang, The existence and asymptotic behavior of solutions to fractional stochastic evolution equations with infinite delay, J. Differential Equations, 266 (2019), 3514-3558.  doi: 10.1016/j.jde.2018.09.009.
    [16] Y. Liu, Y. Wang and T. Caraballo, The continuity, regularity and polynomial stability of mild solutions for stochastic 2D-Stokes equations with unbounded delay driven by tempered fractional Gaussian noise, Stoch. Dyn., (2022). doi: 10.1142/S0219493722500228.
    [17] Y. Liu, Y. Wang and T. Caraballo, Nontrivial equilibrium solutions and general stability for stochastic evolution equations with pantograph delay and tempered fractional noise, (submitted).
    [18] F. Mainardi, On the initial value problem for the fractional diffusion-wave equation, Ser. Adv. Math. Appl. Sci., 23 (1994), 246-251.
    [19] F. MainardiFractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, Imperial College Press, London, 2010.  doi: 10.1142/9781848163300.
    [20] M. M. Meerschaert and F. Sabzikar, Stochastic integration for tempered fractional Brownian motion, Stochastic Process. Appl., 124 (2014), 2363-2387. doi: 10.1016/j.spa.2014.03.002.
    [21] M. M. Meerschaert and A. Sikorskii, Stochastic Models for Fractional Calculus, De Gruyter Studies in Mathematics, 43, 2012.
    [22] Y. S. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Processes, Lecture Notes in Mathematics, 1929. Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-75873-0.
    [23] D. X. Nie and W. H. Deng, A unified convergence analysis for the fractional diffusion equation driven by fractional Gaussion noise with Hurst index $H \in (0, 1)$, arXiv: 2104.13676.
    [24] D. J. Norton and C. V. Wolff, Mobile offshore platform wind loads, Offshore Techn. Conf., OTC 4123, 4 (1981), 77-88. doi: 10.4043/4123-MS.
    [25] I. PodlubnyFractional Differential Equations, Academic Press, London, 1999. 
    [26] A. Shahnazi-Pour, B. P. Moghaddam and A. Babaei, Numerical simulation of the Hurst index of solutions of fractional stochastic dynamical systems driven by fractional Brownian motion, J. Comput. Appl. Math., 386 (2021), Paper No. 113210, 13 pp. doi: 10.1016/j.cam.2020.113210.
    [27] I. M. Sokolov and R. Metzler, Towards deterministic equations for Lévy walks: The fractional material derivative, Phys. Rev. E, 67 (2003), 010101.  doi: 10.1103/PhysRevE.67.010101.
    [28] D. T. SonP. T. HuongP. E. Kloeden and H. T. Tuan, Asymptotic separation between solutions of Caputo fractional stochastic differential equations, Stoch. Anal. Appl., 36 (2018), 654-664.  doi: 10.1080/07362994.2018.1440243.
    [29] R.-N. WangD.-H. Chen and T.-J. Xiao, Abstract fractional Cauchy problems with almost sectorial operators, J. Differential Equations, 252 (2012), 202-235.  doi: 10.1016/j.jde.2011.08.048.
    [30] Y. Wang and T. Liang, Mild solutions to the time fractional Navier-Stokes delay differential inclusions, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 3713-3740.  doi: 10.3934/dcdsb.2018312.
    [31] Y. WangY. Liu and T. Caraballo, Exponential behavior and upper noise excitation index of solutions to evolution equations with unbounded delay and tempered fractional Brownian motions, J. Evol. Equ., 21 (2021), 1779-1807.  doi: 10.1007/s00028-020-00656-0.
    [32] Y. WangJ. Xu and P. E. Kloeden, Asymptotic behavior of stochastic lattice systems with a Caputo fractional time derivative, Nonlinear Anal., 135 (2016), 205-222.  doi: 10.1016/j.na.2016.01.020.
    [33] G. Xiao and J. R. Wang, Stability of solutions of Caputo fractional stochastic differential equations, Nonlinear Anal. Model. Control, 26 (2021), 581-596.  doi: 10.15388/namc.2021.26.22421.
    [34] P. XuG.-A. Zou and J. Huang, Time-space fractional stochastic Ginzburg-Landau equation driven by fractional Brownian motion, Comput. Math. Appl., 78 (2019), 3790-3806.  doi: 10.1016/j.camwa.2019.06.004.
    [35] L. Yan and X. Yin, Optimal error estimates for fractional stochastic partial differential equation with fractional Brownian motion, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 615-635.  doi: 10.3934/dcdsb.2018199.
  • 加载中
SHARE

Article Metrics

HTML views(183) PDF downloads(125) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return