The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion. (English) Zbl 1218.60053

Caraballo, T.; Garrido-Atienza, M. J.; Taniguchi, T.

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74, No. 11, 3671-3684 (2011).

Summary: We investigate the existence, uniqueness and exponential asymptotic behavior of mild solutions to stochastic delay evolution equations perturbed by a fractional Brownian motion \(B^H_Q(t)\),
\[ dX(t)=(AX(t)+f(t,X_t))\,dt+g(t)\,dB^H_Q(t), \]
with Hurst parameter \(H\in (1/2,1)\). We also consider the existence of weak solutions.

MSC:

60H15	Stochastic partial differential equations (aspects of stochastic analysis)
60G22	Fractional processes, including fractional Brownian motion

Keywords:

delay stochastic PDEs; fractional Brownian motion; exponential decay in mean square

References:

[1]	Nualart, D.; Răşcanu, A., Differential equations driven by fractional Brownian motion, Collect. Math., 53, 1, 55-81 (2002) · Zbl 1018.60057
[2]	Garrido-Atienza, M. J.; Maslowski, B.; Schmalfuss, B., Random attractors for stochastic equations driven by a fractional Brownian motion, Int. J. Bifur. Chaos, 20, 9, 2761-2782 (2010) · Zbl 1202.37073
[3]	Grecksch, W.; Anh, V. V., A parabolic stochastic differential equation with fractional Brownian motion input, Statist. Probab. Lett., 41, 337-345 (1999) · Zbl 0937.60064
[4]	Maslowski, B.; Nualart, D., Evolution equations driven by a fractional Brownian motion, J. Funct. Anal., 202, 1, 277-305 (2003) · Zbl 1027.60060
[5]	Tindel, S.; Tudor, C.; Viens, F., Stochastic evolution equations with fractional brownian motion, Probab. Theory Related Fields, 127, no. 2, 186-204 (2003) · Zbl 1036.60056
[6]	Gubinelli, M.; Lejay, A.; Tindel, S., Young integrals and SPDEs, Potential Anal., 25, 4, 307-326 (2006) · Zbl 1103.60062
[7]	Garrido-Atienza, M. J.; Lu, K.; Schmalfuss, B., Random dynamical systems for stochastic partial differential equations driven by a fractional Brownian motion, Discrete Contin. Dyn. Syst. Ser. B, 14, 2, 473-493 (2010) · Zbl 1200.37075
[8]	Hale, J. K.; Verduyn Lunel, S. M., Introduction to Functional Differential Equations (1995), Springer: Springer New York
[9]	Manitius, A., Feedback controllers for a wind tunnel model involving a delay: analytical design and numerical simulation, IEEE Trans. Automat. Control, 29, 12, 1058-1068 (1984)
[10]	Kuang, Y., Delay Differential Equations with Applications in Population Dynamics (1993), Academic Press: Academic Press Boston · Zbl 0777.34002
[11]	Murray, J. D., Mathematical Biology (1993), Springer-Verlag: Springer-Verlag Berlin · Zbl 0779.92001
[12]	Ferrante, M.; Rovira, C., Stochastic delay differential equations driven by fractional Brownian motion with Hurst parameter \(H > 1 / 2\), Bernouilli, 12, 85-100 (2006) · Zbl 1102.60054
[13]	Neuenkirch, A.; Nourdin, I.; Tindel, S., Delay equations driven by rough paths, Electron. J. Probab., 13, 2031-2068 (2008) · Zbl 1190.60046
[14]	Ferrante, M.; Rovira, C., Convergence of delay differential equations driven by fractional Brownian motion, J. Evol. Equ., 10, 4, 761-783 (2010) · Zbl 1239.60040
[16]	Alos, E.; Mazet, O.; Nualart, D., Stochastic calculus with respect to Gaussian processes, Ann. Probab., 29, 766-801 (1999) · Zbl 1015.60047
[17]	Mishura, Y., (Stochastic Calculus for Fractional Brownian Motion and Related Topics. Stochastic Calculus for Fractional Brownian Motion and Related Topics, Lecture Notes in Mathematics, 1929 (2008))
[18]	Nualart, D., The Malliavin Calculus and Related Topics (2006), Springer-Verlag: Springer-Verlag Berlin · Zbl 1099.60003
[19]	Da Prato, G.; Zabczyk, J., Stochastic Equations in Infinite Dimensions (1992), Cambridge University Press · Zbl 0761.60052