# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.