March  2020, 19(3): 1291-1319. doi: 10.3934/cpaa.2020063

Averaging principle for stochastic real Ginzburg-Landau equation driven by $ \alpha $-stable process

1. 

School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, 221116, China

2. 

Key Laboratory of Wu Wen-Tsun Mathematics, CAS, School of Mathematical Science, University of Science and Technology of China, Hefei, 230026, China

* Corresponding author

Received  March 2019 Revised  August 2019 Published  November 2019

Fund Project: Xiaobin Sun is supported by the National Natural Science Foundation of China (11601196, 11771187, 11931004), the NSF of Jiangsu Province (No. BK20160004) and the Priority Academic Program Development of Jiangsu Higher Education Institutions. Jianliang Zhai is supported by the National Natural Science Foundation of China (11431014, 11671372, 11721101), the Fundamental Research Funds for the Central Universities (No. WK0010450002, WK3470000008), Key Research Program of Frontier Sciences, CAS, No: QYZDB-SSW-SYS009, School Start-up Fund (USTC) KY0010000036

In this paper, we study a system of stochastic partial differential equations with slow and fast time-scales, where the slow component is a stochastic real Ginzburg-Landau equation and the fast component is a stochastic reaction-diffusion equation, the system is driven by cylindrical $ \alpha $-stable process with $ \alpha\in (1, 2) $. Using the classical Khasminskii approach based on time discretization and the techniques of stopping times, we show that the slow component strong converges to the solution of the corresponding averaged equation under some suitable conditions.

Citation: Xiaobin Sun, Jianliang Zhai. Averaging principle for stochastic real Ginzburg-Landau equation driven by $ \alpha $-stable process. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1291-1319. doi: 10.3934/cpaa.2020063
References:
[1]

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W. Liu, M. Röckner, X. Sun and Y. Xie, Averaging principle for slow-fast stochastic differential equations with time dependent locally Lipschitz coefficients, J. Differential Equations, (2019). doi: 10.1016/j.jde.2019.09.047.  Google Scholar

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show all references

References:
[1]

J. BaoG. Yin and C. Yuan, Two-time-scale stochastic partial differential equations driven by $\alpha$-stable noises: averaging principles, Bernoulli, 23 (2017), 645-669.  doi: 10.3150/14-BEJ677.  Google Scholar

[2]

N. N. Bogoliubov and Y. A. Mitropolsk, Asymptotic Methods in the Theory of Non-linear Oscillations, Gordon and Breach Science Publishers, New York, 1961.  Google Scholar

[3]

C. E. Bréhier, Strong and weak orders in averaging for SPDEs, Stochastic Process. Appl., 122 (2012), 2553-2593.  doi: 10.1016/j.spa.2012.04.007.  Google Scholar

[4]

S. Cerrai, A Khasminskii type averaging principle for stochastic reaction-diffusion equations, Ann. Appl. Probab., 19 (2009), 899-948.  doi: 10.1214/08-AAP560.  Google Scholar

[5]

S. Cerrai, Averaging principle for systems of reaction-diffusion equations with polynomial nonlinearities perturbed by multiplicative noise, SIAM J. Math. Anal., 43 (2011), 2482-2518.  doi: 10.1137/100806710.  Google Scholar

[6]

S. Cerrai and M. Freidli., Averaging principle for stochastic reaction-diffusion equations, Probab. Theory Related Fields, 144 (2009), 137-177.  doi: 10.1007/s00440-008-0144-z.  Google Scholar

[7]

Z. DongX. SunH. Xiao and J. Zhai, Averaging principle for one dimensional stochastic Burgers equation, J. Differential Equations, 265 (2018), 4749-4797.  doi: 10.1016/j.jde.2018.06.020.  Google Scholar

[8]

Z. DongL. Xu and X. Zhang, Invariance measures of stochastic 2D Navier-stokes equations driven by $\alpha$-stable processes, Electronic Communications in Probability, 16 (2011), 678-688.  doi: 10.1214/ECP.v16-1664.  Google Scholar

[9]

Z. DongL. Xu and X. Zhang, Exponential ergodicity of stochastic Burgers equations driven by $\alpha$-stable processes, J. Stat. Phys., 154 (2014), 929-949.  doi: 10.1007/s10955-013-0881-y.  Google Scholar

[10]

H. Fu and J. Liu, Strong convergence in stochastic averaging principle for two time-scales stochastic partial differential equations, J. Math. Anal. Appl., 384 (2011), 70-86.  doi: 10.1016/j.jmaa.2011.02.076.  Google Scholar

[11]

H. FuL. Wan and J. Liu, Strong convergence in averaging principle for stochastic hyperbolic-parabolic equations with two time-scales, Stochastic Process. Appl., 125 (2015), 3255-3279.  doi: 10.1016/j.spa.2015.03.004.  Google Scholar

[12]

H. FuL. WanY. Wang and J. Liu, Strong convergence rate in averaging principle for stochastic FitzHugh-Nagumo system with two time-scales, J. Math. Anal. Appl., 416 (2014), 609-628.  doi: 10.1016/j.jmaa.2014.02.062.  Google Scholar

[13]

P. Gao, Averaging principle for stochastic Kuramoto-Sivashinsky equation with a fast oscillation, Discrete Contin. Dyn. Syst.-A, 38 (2018), 5649-5684.  doi: 10.3934/dcds.2018247.  Google Scholar

[14]

P. Gao, Averaging principle for the higher order nonlinear Schrödinger equation with a random fast oscillation, J. Stat. Phys., 171 (2018), 897-926.  doi: 10.1007/s10955-018-2048-3.  Google Scholar

[15]

P. Gao, Averaging principle for multiscale stochastic Klein-Gordon-Heat system, J Nonlinear Sci., 29 (2019), 1701-1759.  doi: 10.1007/s00332-019-09529-4.  Google Scholar

[16]

D. Givon, Strong convergence rate for two-time-scale jump-diffusion stochastic differential systems, Multiscale Model. Simul., 6 (2007), 577-594.  doi: 10.1137/060673345.  Google Scholar

[17]

D. GivonI. G. Kevrekidis and R. Kupferman, Strong convergence of projective integeration schemes for singularly perturbed stochastic differential systems, Comm. Math. Sci., 4 (2006), 707-729.   Google Scholar

[18]

J. Golec, Stochastic averaging principle for systems with pathwise uniqueness, Stochastic Anal. Appl., 13 (1995), 307-322.  doi: 10.1080/07362999508809400.  Google Scholar

[19]

J. Golec and G. Ladde, Averaging principle and systems of singularly perturbed stochastic differential equations, J. Math. Phys., 31 (1990), 1116-1123.  doi: 10.1063/1.528792.  Google Scholar

[20]

A. Ichikawa, Some inequalities for martingales and stochastic convolutions, Stoch. Anal. Appl., 4 (1986), 329-339.  doi: 10.1080/07362998608809094.  Google Scholar

[21]

S. Li, X. Sun, Y. Xie and Y. Zhao, Averaging principle for two dimensional stochatsic Navier-Stokes equations, arXiv: 1810.02282. Google Scholar

[22]

D. Liu, Strong convergence of principle of averaging for multiscale stochastic dynamical systems, Commun. Math. Sci., 8 (2010), 999-1020.   Google Scholar

[23]

W. Liu, M. Röckner, X. Sun and Y. Xie, Averaging principle for slow-fast stochastic differential equations with time dependent locally Lipschitz coefficients, J. Differential Equations, (2019). doi: 10.1016/j.jde.2019.09.047.  Google Scholar

[24]

Y. Liu and J. Zhai, A note on time regularity of generalized Ornstein-Uhlenbeck processes with cylindrical stable noise, C. R. Math. Acad. Sci. Paris, 350 (2012), 97-100.  doi: 10.1016/j.crma.2011.11.017.  Google Scholar

[25]

R. Z. Khasminskii, On an averaging principle for Itô stochastic differential equations, Kibernetica, 4 (1968), 260-279.   Google Scholar

[26]

S. X. Ouyang, Harnack Inequalities and Applications for Stochastic Equations, Ph.D thesis, Bielefeld University, 2019. Google Scholar

[27]

B. PeiY. Xu and G. Yin, Stochastic averaging for a class of two-time-scale systems of stochastic partial differential equations, Nonlinear Anal., 160 (2017), 159-176.  doi: 10.1016/j.na.2017.05.005.  Google Scholar

[28]

E. Priola and J. Zabczyk, Structural properties of semilinear SPDEs driven by cylindrical stable processes, Probability Theory and Related Fields, 149 (2011), 97-137.  doi: 10.1007/s00440-009-0243-5.  Google Scholar

[29]

K. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge Studies in Advanced Mathematics, vol. 68, Cambridge University Press, Cambridge, 1999.  Google Scholar

[30]

F. Y. Wang, Gradient estimate for Ornstein-Uhlenbeck jump processes, Stochastic Process. Appl., 121 (2011), 466-478.  doi: 10.1016/j.spa.2010.12.002.  Google Scholar

[31]

W. Wang and A. J. Roberts, Average and deviation for slow-fast stochastic partial differential equations, J.Differential Equations, 253 (2012), 1265-1286.  doi: 10.1016/j.jde.2012.05.011.  Google Scholar

[32]

W. WangA. J. Roberts and J. Duan, Large deviations and approximations for slow-fast stochastic reaction-diffusion equations, J.Differential Equations, 253 (2012), 3501-3522.  doi: 10.1016/j.jde.2012.08.041.  Google Scholar

[33]

J. XuY. Miao and J. Liu, Strong averaging principle for slow-fast SPDEs with Poisson random measures, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2233-2256.  doi: 10.3934/dcdsb.2015.20.2233.  Google Scholar

[34]

L. Xu, Ergodicity of the stochastic real Ginzburg-Landau equation driven by $\alpha$-stable noises, Stochastic Process. Appl., 123 (2013), 3710-3736.  doi: 10.1016/j.spa.2013.05.002.  Google Scholar

[35]

Y. Xu, B. Pei and J.-L. Wu, Stochastic averaging principle for differential equations with non-Lipschitz coefficients driven by fractional Brownian motion, Stoch. Dyn., 17 (2017), 1750013. doi: 10.1142/S0219493717500137.  Google Scholar

[36]

X. Zhang, Derivative formulas and gradient estimates for SDEs driven by $\alpha$-stable processes, Stochastic Process. Appl., 123 (2013), 1213-1228.  doi: 10.1016/j.spa.2012.11.012.  Google Scholar

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