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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[18]

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

[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.

[20]

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

[21]

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

[22]

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

[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.

[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.

[25]

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

[26]

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

[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.

[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.

[29]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[18]

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

[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.

[20]

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

[21]

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

[22]

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

[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.

[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.

[25]

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

[26]

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

[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.

[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.

[29]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[1]

Tianlong Shen, Jianhua Huang. Ergodicity of the stochastic coupled fractional Ginzburg-Landau equations driven by α-stable noise. Discrete and Continuous Dynamical Systems - B, 2017, 22 (2) : 605-625. doi: 10.3934/dcdsb.2017029

[2]

Yan Zheng, Jianhua Huang. Exponential convergence for the 3D stochastic cubic Ginzburg-Landau equation with degenerate noise. Discrete and Continuous Dynamical Systems - B, 2019, 24 (10) : 5621-5632. doi: 10.3934/dcdsb.2019075

[3]

Noboru Okazawa, Tomomi Yokota. Subdifferential operator approach to strong wellposedness of the complex Ginzburg-Landau equation. Discrete and Continuous Dynamical Systems, 2010, 28 (1) : 311-341. doi: 10.3934/dcds.2010.28.311

[4]

Feng Zhou, Chunyou Sun. Dynamics for the complex Ginzburg-Landau equation on non-cylindrical domains I: The diffeomorphism case. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3767-3792. doi: 10.3934/dcdsb.2016120

[5]

Shujuan Lü, Hong Lu, Zhaosheng Feng. Stochastic dynamics of 2D fractional Ginzburg-Landau equation with multiplicative noise. Discrete and Continuous Dynamical Systems - B, 2016, 21 (2) : 575-590. doi: 10.3934/dcdsb.2016.21.575

[6]

Lu Zhang, Aihong Zou, Tao Yan, Ji Shu. Weak pullback attractors for stochastic Ginzburg-Landau equations in Bochner spaces. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 749-768. doi: 10.3934/dcdsb.2021063

[7]

N. Maaroufi. Topological entropy by unit length for the Ginzburg-Landau equation on the line. Discrete and Continuous Dynamical Systems, 2014, 34 (2) : 647-662. doi: 10.3934/dcds.2014.34.647

[8]

Hans G. Kaper, Peter Takáč. Bifurcating vortex solutions of the complex Ginzburg-Landau equation. Discrete and Continuous Dynamical Systems, 1999, 5 (4) : 871-880. doi: 10.3934/dcds.1999.5.871

[9]

Jingna Li, Li Xia. The Fractional Ginzburg-Landau equation with distributional initial data. Communications on Pure and Applied Analysis, 2013, 12 (5) : 2173-2187. doi: 10.3934/cpaa.2013.12.2173

[10]

Satoshi Kosugi, Yoshihisa Morita, Shoji Yotsutani. A complete bifurcation diagram of the Ginzburg-Landau equation with periodic boundary conditions. Communications on Pure and Applied Analysis, 2005, 4 (3) : 665-682. doi: 10.3934/cpaa.2005.4.665

[11]

Jun Yang. Vortex structures for Klein-Gordon equation with Ginzburg-Landau nonlinearity. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 2359-2388. doi: 10.3934/dcds.2014.34.2359

[12]

Sen-Zhong Huang, Peter Takáč. Global smooth solutions of the complex Ginzburg-Landau equation and their dynamical properties. Discrete and Continuous Dynamical Systems, 1999, 5 (4) : 825-848. doi: 10.3934/dcds.1999.5.825

[13]

Simão Correia, Mário Figueira. A generalized complex Ginzburg-Landau equation: Global existence and stability results. Communications on Pure and Applied Analysis, 2021, 20 (5) : 2021-2038. doi: 10.3934/cpaa.2021056

[14]

Francisco Guillén-González, Mouhamadou Samsidy Goudiaby. Stability and convergence at infinite time of several fully discrete schemes for a Ginzburg-Landau model for nematic liquid crystal flows. Discrete and Continuous Dynamical Systems, 2012, 32 (12) : 4229-4246. doi: 10.3934/dcds.2012.32.4229

[15]

Dingshi Li, Xiaohu Wang. Asymptotic behavior of stochastic complex Ginzburg-Landau equations with deterministic non-autonomous forcing on thin domains. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 449-465. doi: 10.3934/dcdsb.2018181

[16]

Dingshi Li, Lin Shi, Xiaohu Wang. Long term behavior of stochastic discrete complex Ginzburg-Landau equations with time delays in weighted spaces. Discrete and Continuous Dynamical Systems - B, 2019, 24 (9) : 5121-5148. doi: 10.3934/dcdsb.2019046

[17]

Dandan Ma, Ji Shu, Ling Qin. Wong-Zakai approximations and asymptotic behavior of stochastic Ginzburg-Landau equations. Discrete and Continuous Dynamical Systems - B, 2020, 25 (11) : 4335-4359. doi: 10.3934/dcdsb.2020100

[18]

Yun Lan, Ji Shu. Dynamics of non-autonomous fractional stochastic Ginzburg-Landau equations with multiplicative noise. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2409-2431. doi: 10.3934/cpaa.2019109

[19]

Hans G. Kaper, Bixiang Wang, Shouhong Wang. Determining nodes for the Ginzburg-Landau equations of superconductivity. Discrete and Continuous Dynamical Systems, 1998, 4 (2) : 205-224. doi: 10.3934/dcds.1998.4.205

[20]

Mickaël Dos Santos, Oleksandr Misiats. Ginzburg-Landau model with small pinning domains. Networks and Heterogeneous Media, 2011, 6 (4) : 715-753. doi: 10.3934/nhm.2011.6.715

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (248)
  • HTML views (81)
  • Cited by (4)

Other articles
by authors

[Back to Top]