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Stochastic averaging for two-time-scale stochastic partial differential equations with fractional Brownian motion. (English) Zbl 1418.60083
Summary: In this paper, we are concerned with a class of stochastic partial differential equations that have a slow component driven by a fractional Brownian motion with Hurst parameter \(0 < H < 1 / 2\) and a fast component driven by a fast-varying diffusion. We will establish an averaging principle in which the fast-varying diffusion process acts as a “noise” and is averaged out in the limit. The slow process is shown to have a limit in the \(L^2\) sense, which is characterized by the solution to a stochastic partial differential equation driven by a fractional Brownian motion with Hurst parameter \(0 < H < 1 / 2\) whose coefficients are averages of that of the original slow process with respect to the stationary measure of the fast-varying diffusion. In the end, one example is given to illustrate the feasibility and effectiveness of results obtained.

MSC:
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60G15 Gaussian processes
60H05 Stochastic integrals
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[1] Bogoliubov, N. N.; Mitropolsky, Y. A., Asymptotic Methods in the Theory of Non-linear Oscillations, (1961), Gordon and Breach Science Publishers: Gordon and Breach Science Publishers New York
[2] Besjes, J. G., On the asymptotic methods for non-linear differential equations, J. Mec., 8, 357-373, (1969) · Zbl 0238.34074
[3] Khasminskii, R., A limit theorem for the solutions of differential equations with random right-hand sides, Theory Probab. Appl., 11, 3, 390-406, (1966)
[4] Khasminskii, R. Z., On the principle of averaging the Itô stochastic differential equations, Kibernet, 4, 260-279, (1968), (in Russian) · Zbl 0231.60045
[5] Sri. Namachchivaya, N.; Lin, Y. K., Application of stochastic averaging for systems with high damping, Probab. Eng. Mech., 3, 185-196, (1988)
[6] Roberts, J.; Spanos, P., Stochastic averaging: an approximate method of solving random vibration problems, Int. J. Non-Linear Mech., 21, 111-134, (1986) · Zbl 0582.73077
[7] Liptser, R.; Spokoiny, V., On estimating a dynamic function of a stochastic system with averaging, Stat. Inference Stoch. Process., 3, 225-249, (2000) · Zbl 0982.62070
[8] Cerrai, S.; Freidlin, M., Averaging principle for a class of stochastic reaction diffusion equations, Probab. Theory Related Fields, 144, 1-2, 137-177, (2009) · Zbl 1176.60049
[9] Cerrai, S., A khasminskii type averaging principle for stochastic reaction-diffusion equations, Ann. Appl. Probab., 19, 3, 899-948, (2009) · Zbl 1191.60076
[10] Bréhier, C. E., Strong and weak orders in averaging for SPDEs, Stochastic Process. Appl., 122, 7, 2553-2593, (2012) · Zbl 1266.60112
[11] Xu, J.; Miao, Y.; Liu, J., Strong averaging principle for slow-fast spdes with Poisson random measures, Discrete Contin. Dyn. Syst. Ser. B, 20, 7, 2233-2256, (2015) · Zbl 1335.60118
[12] Xu, Y.; Pei, B.; Li, Y., Approximation properties for solutions to non-Lipschitz stochastic differential equations with Lévy noise, Math. Methods Appl. Sci., 38, 11, 2120-2131, (2015) · Zbl 1345.60051
[13] Thompson, W. F.; Kuske, R. A.; Monahan, A. H., Stochastic averaging of dynamical systems with multiple time scales forced with \(\alpha\)-stable noise, Multiscale Model. Simul., 13, 4, 1194-1223, (2015) · Zbl 1333.34099
[14] Bao, J.; Yin, G.; Yuan, C., Two-time-scale stochastic partial differential equations driven by \(\alpha\)-stable noises: Averaging principles, Bernoulli, 23, 1, 645-669, (2017) · Zbl 1360.60118
[15] Comte, F.; Renault, E., Long memory continuous time models, J. Econometrics, 73, 101-149, (1996) · Zbl 0856.62104
[16] De La, F.; Perez-Samartin, A. L.; Matnez, L.; Garcia, M. A.; Vera-Lopez, A., Long-range correlations in rabbit brain neural activity, Ann. Biomed. Eng., 34, 295-299, (2006)
[17] Willinger, W.; Leland, W.; Taqqu, M.; Wilson, D., On self-similar nature of ethernet traffic, IEEE/ACM Trans. Netw., 2, 1-15, (1994)
[18] Rypdal, M.; Rypdal, K., Testing hypotheses about sun-climate complexity linking, Phys. Rev. Lett., 104, 128-151, (2010)
[19] Simonsen, I., Measuring anti-correlations in the nordic electricity spot market by wavelets, Physica A, 322, 597-606, (2003) · Zbl 1017.91026
[20] Boufoussi, B.; Hajji, S., Neutral stochastic functional differential equations driven by a fractional Brownian motion in a Hilbert space, Statist. Probab. Lett., 82, 8, 1549-1558, (2012) · Zbl 1248.60069
[21] Caraballo, T.; Garrido-Atienza, M. J.; Taniguchi, T., The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion, Nonlinear Anal., 74, 3671-3684, (2011) · Zbl 1218.60053
[22] Duncan, T. E.; Maslowski, B.; Pasik-Duncan, B., Fractional Brownian motion and stochastic equations in Hilbert spaces, Stoch. Dyn., 2, 225-250, (2002) · Zbl 1040.60054
[23] Maslowski, B.; Nualart, D., Evolution equations driven by a fractional Brownian motion, J. Funct. Anal., 202, 277-305, (2003) · Zbl 1027.60060
[24] Ren, Y.; Chen, X.; Sakthivel, R., Impulsive neutral stochastic functional integro-differential equations with infinite delay driven by fBm, Appl. Math. Comput., 247, 15, 205-212, (2014) · Zbl 1338.34157
[25] Xu, Y.; Guo, R.; Liu, D.; Zhang, H. Q.; Duan, J. Q., Stochastic averaging principle for dynamical systems with fractional Brownian motion, Discrete Contin. Dyn. Syst. Ser. B, 19, 4, 1197-1212, (2014) · Zbl 1314.60122
[26] Xu, Y.; Pin, B.; Li, Y. G., An averaging principle for stochastic differential delay equations with fractional Brownian motion, Abstr. Appl. Anal., (2014)
[27] Xu, Y.; Pin, B.; Guo, R., Stochastic averaging for slow-fast dynamical systems with fractional Brownian motion, Discrete Contin. Dyn. Syst. Ser. B, 20, 7, 2257-2267, (2015) · Zbl 1335.34090
[28] Xu, Y.; Pin, B.; Wu, J. L., Stochastic averaging principle for differential equations with non-Lipschitz coefficients driven by fractional Brownian motion, Stoch. Dyn., 17, 2, (2017)
[29] Pei, B.; Xu, Y.; Yin, G.; Zhang, X. Y., Averaging principles for functional stochastic partial differential equations driven by a fractional Brownian motion modulated by two-time-scale Markovian switching processes, Nonlinear Anal. Hybrid Syst., 27, 107-124, (2018) · Zbl 1380.60060
[30] Pei, B.; Xu, Y.; Yin, G., Averaging principles for SPDEs driven by fractional Brownian motions with random delays modulated by two-time-scale Markov switching processes, Stoch. Dyn., (2017)
[31] 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., 195, 145-152, (2009)
[32] Pei, B.; Xu, Y.; Yin, G., Stochastic averaging for a class of two-time-scale systems of stochastic partial differential equations, Nonlinear Anal., 160, 159-176, (2017) · Zbl 1370.60108
[33] Biagini, F.; Hu, Y. Z.; Øksendal, B.; Zhang, T. S., Stochastic Calculus for Fractional Brownian Motion and Applications, (2008), Springer-Verlag London
[34] Nualart, D., The Malliavin Calculus and Related Topics, (2006), Springer-Verlag: Springer-Verlag Berlin · Zbl 1099.60003
[35] Pazy, A., Semigroup of Linear Operators and Applications to Partial Differential Equations, (1992), Spring Verlag: Spring Verlag New York · Zbl 0516.47023
[36] Fu, H.; Liu, J., Strong convergence in stochastic averaging principle for two time-scales stochastic partial differential equations, J. Math. Anal. Appl., 384, 1, 70-86, (2011) · Zbl 1223.60044
[37] Fu, H.; Wan, L.; Liu, J., Strong convergence in averaging principle for stochastic hyperbolic-parabolic equations with two time-scales, Stochastic Process. Appl., 125, 8, 3255-3279, (2015) · Zbl 1322.60111
[38] Chow, P. L., Stochastic Partial Differential Equations, (2007), Chapman & Hall/CRC: Chapman & Hall/CRC New York
[39] Øksendal, B., Stochstic Differential Equations, (2003), Springer
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