Wong-Zakai approximations and asymptotic behavior of stochastic Ginzburg-Landau equations

Dandan Ma; Ji Shu; Ling Qin

doi:10.3934/dcdsb.2020100

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November 2020, 25(11): 4335-4359. doi: 10.3934/dcdsb.2020100

Wong-Zakai approximations and asymptotic behavior of stochastic Ginzburg-Landau equations

Dandan Ma , Ji Shu ^, and Ling Qin

School of Mathematical Science and V.C. & V.R. Key Lab, Sichuan Normal University, Chengdu, Sichuan 610068, China

^* Corresponding author: Ji Shu, shuji@sicnu.edu.cn

Received August 2011 Published April 2020

Fund Project: The second author is supported by NSFC (11871138), joint research project of Laurent Mathematics Center of Sichuan Normal University and National-Local Joint Engineering Laboratory of System Credibility Automatic Verification and the funding of V. C. & V. R. Key Lab of Sichuan Province.

In this paper we discuss the Wong-Zakai approximations given by a stationary process via the Wiener shift and their associated long term pathwise behavior for stochastic Ginzburg-Landau equations driven by a white noise. We first apply the Galerkin method and compactness argument to prove the existence and uniqueness of weak solutions. Consequently, we show that the approximate equation has a pullback random attractor under much weaker conditions than the original stochastic equation. At last, when the stochastic Ginzburg-Landau equation is driven by a linear multiplicative noise, we establish the convergence of solutions of Wong-Zakai approximations and the upper semicontinuity of random attractors of the approximate random system as the size of approximation approaches zero.

Keywords: Wong-Zakai approximations, Random attractor, Ginzburg-Landau equation, Upper semicontinuity, White noise.

Mathematics Subject Classification: Primary: 37L55, 60H15; Secondary: 35Q56.

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

References:

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[2]	V. Bally, A. Millet and M. Sanz-Solé, Approximation and support theorem in Hölder norm for parabolic stochastic partial differential equations, Ann. Probab., 23 (1995), 178-222. doi: 10.1214/aop/1176988383. Google Scholar
[3]	M. Bartuccelli, P. Constantin, C. R. Doering, J. Gibbon and M. Gisselfält, On the possibility of soft and hard turbulence in the complex Ginzburg-Landau equation, Phys. D, 44 (1990), 421-444. doi: 10.1016/0167-2789(90)90156-J. Google Scholar
[4]	Z. Brzeźniak, M. Capiński and F. Flandoli, A convergence result for stochastic partial differential equations, Stochastics, 24 (1988), 423-445. doi: 10.1080/17442508808833526. Google Scholar
[5]	Z. Brzeźniak and F. Flandoli, Almost sure approximation of Wong-Zakai type for stochastic partial differential equations, Stochastic Process. Appl., 55 (1995), 329-358. doi: 10.1016/0304-4149(94)00037-T. Google Scholar
[6]	T. Caraballo, J. A. Langa, V. S. Melnik and J. Valero, Pullback attractors for nonautonomous and stochastic multivalued dynamical systems, Set-Valued Anal., 11 (2003), 153-201. doi: 10.1023/A:1022902802385. Google Scholar
[7]	H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393. doi: 10.1007/BF01193705. Google Scholar
[8]	A. Deya, M. Jolis and L. Quer-Sardanyons, The Stratonovich heat equation: A continuity result and weak approximations, Electron. J. Probab., 18 (2013), 34 pp. doi: 10.1214/EJP.v18-2004. Google Scholar
[9]	C. R. Doering, J. D. Gibbon and C. D. Levermore, Weak and strong solutions of the complex Ginzburg-Landau equation, Phys. D, 71 (1994), 285-318. doi: 10.1016/0167-2789(94)90150-3. Google Scholar
[10]	J. Duan, P. Holmes and E. S. Titi, Global existence theory for a generalized Ginzburg-Landau equation, Nonlinearity, 5 (1992), 1303-1314. doi: 10.1088/0951-7715/5/6/005. Google Scholar
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[21]	M. Hairer and E. Pardoux, A Wong-Zakai theorem for stochastic PDEs, J. Math. Soc. Japan, 67 (2015), 1551-1604. doi: 10.2969/jmsj/06741551. Google Scholar
[22]	N. Ikeda, S. Nakao and Y. Yamato, A class of approximations of Brownian motion, Publ. Res. Inst. Math. Sci., 13 (1977/78), 285-300. doi: 10.2977/prims/1195190109. Google Scholar
[23]	N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, 2$^nd$ edition, North-Holland Publishing Co., Amsterdam, Kodansha, Ltd., Tokyo, 1989. Google Scholar
[24]	D. Kelley and I. Melbourne, Smooth approximation of stochastic differential equations, Ann. Probab., 44 (2016), 479-520. doi: 10.1214/14-AOP979. Google Scholar
[25]	F. Konecny, On Wong-Zakai approximation of stochastic differential equations, J. Multivariate Anal., 13 (1983), 605-611. doi: 10.1016/0047-259X(83)90043-X. Google Scholar
[26]	T. G. Kurtz and P. Protter, Weak limit theorems for stochastic integrals and stochastic differential equations, Ann. Probab., 19 (1991), 1035-1070. doi: 10.1214/aop/1176990334. Google Scholar
[27]	T. Kurtz and P. Protter, Wong-Zakai corrections, random evolutions, and simulation schemes for SDE, in Stochastic Analysis: Liber Amicorum for Moshe Zakai, Academic Press, Boston, (1991), 331–346. doi: 10.1016/B978-0-12-481005-1.50023-5. Google Scholar
[28]	Y. Lan and J. Shu, Dynamics of non-autonomous fractional stochastic Ginzburg-Landau equations with multiplicative noise, Commun. Pure Appl. Anal., 18 (2019), 2409-2431. doi: 10.3934/cpaa.2019109. Google Scholar
[29]	Y. Lan and J. Shu, Fractal dimension of random attractors for non-autonomous fractional stochastic Ginzburg-Landau equations with multiplicative noise, Dyn. Syst., 34 (2019), 274-300. doi: 10.1080/14689367.2018.1523368. Google Scholar
[30]	D. Li, Z. Dai and X. Liu, Long time behaviour for generalized complex Ginzburg-Landau equation, J. Math. Anal. Appl., 330 (2007), 934-948. doi: 10.1016/j.jmaa.2006.07.095. Google Scholar
[31]	D. Li and B. Guo, Asymptotic behavior of the 2D generalized stochastic Ginzburg-Landau equation with additive noise, Appl. Math. Mech. (English Ed.), 30 (2009), 945-956. doi: 10.1007/s10483-009-0801-x. Google Scholar
[32]	D. Li, K. Lu, B. Wang and X. Wang, Limiting behavior of dynamics for stochastic reaction-diffusion equations with additive noise on thin domains, Discrete Contin. Dyn. Syst., 38 (2018), 187-208. doi: 10.3934/dcds.2018009. Google Scholar
[33]	D. Li, K. Lu, B. Wang and X. Wang, Limiting dynamics for non-autonomous stochastic retarded reaction-diffusion equations on thin domains, Discrete Contin. Dyn. Syst., 39 (2019), 3717-3747. doi: 10.3934/dcds.2019151. Google Scholar
[34]	J. L. Lions, Quelques Methodes de Resolution des Problemes aux Limites Non Lineaires, Dunod, Gauthier-Villars, Paris, 1969. Google Scholar
[35]	K. Lu, and B. Wang, Wong-Zakai approximations and long term behavior of stochastic partial differential equations, J. Dynam. Differential Equations, (2019), 1341–1371. doi: 10.1007/s10884-017-9626-y. Google Scholar
[36]	K. Lu and Q. Wang, Chaotic behavior in differential equations driven by a Brownian motion, J. Differential Equations, 251 (2011), 2853-2895. doi: 10.1016/j.jde.2011.05.032. Google Scholar
[37]	E. J. McShane, Stochastic Differential Equations and Models of Random Processes. Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Vol. Ⅲ, Univ. California Press, Berkeley, Calif., 1972,263{294. Google Scholar
[38]	S. Nakao, On weak convergence of sequences of continuous local martingales, Ann. Inst. H. Poincar?Probab. Statist., 22 (1986), 371-380. Google Scholar
[39]	S. Nakao and Y. Yamato, Approximation theorem on stochastic differential equations, in Proceedings of the International Symposium on Stochastic Differential Equations, Wiley, New York-Chinchester-Brisbane, 1978,283–296. Google Scholar
[40]	A. Nowak, A Wong-Zakai type theorem for stochastic systems of Burgers equations, PanAmer. Math. J., 16 (2006), 1-25. Google Scholar
[41]	P. Protter, Approximations of solutions of stochastic differential equations driven by semimartingales, Ann. Probab., 13 (1985), 716-743. doi: 10.1214/aop/1176992905. Google Scholar
[42]	B. Schmalfuss, V. Reitmann, T. Riedrich and N. Koksch (eds.), Backward cocycles and attractors of stochastic differential equations, in International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior, Technische Universitat, Dresden, 1992,185–192. Google Scholar
[43]	J. Shen, K. Lu and W. Zhang, Heteroclinic chaotic behavior driven by a Brownian motion, J. Differential Equations, 255 (2013), 4185-4225. doi: 10.1016/j.jde.2013.08.003. Google Scholar
[44]	J. Shu, P. Li, J. Zhang and O. Liao, Random attractors for the stochastic coupled fractional Ginzburg-Landau equation with additive noise, J. Math. Phys., 56 (2015), 102702, 11 pp. doi: 10.1063/1.4934724. Google Scholar
[45]	D. W. Stroock and S. R. S. Varadhan, On the support of diffusion processes with applications to the strong maximum principle, in Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Vol. Ⅲ: Probability Theory, Univ. California Press, Berkeley, Calif., 1972,333{359. Google Scholar
[46]	H. J. Sussmann, An interpretation of stochastic differential equations as ordinary differential equations which depend on the sample point, Bull. Amer. Math. Soc., 83 (1977), 296-298. doi: 10.1090/S0002-9904-1977-14312-7. Google Scholar
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[48]	R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2$^{nd}$ edition, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3. Google Scholar
[49]	G. Tessitore and J. Zabczyk, Wong-Zakai approximations of stochastic evolution equations, J. Evol. Equ., 6 (2006), 621-655. doi: 10.1007/s00028-006-0280-9. Google Scholar
[50]	K. Twardowska, An approximation theorem of Wong-Zakai type for nonlinear stochastic partial differential equations, Stochastic Anal. Appl., 13 (1995), 601-626. doi: 10.1080/07362999508809419. Google Scholar
[51]	K. Twardowska, An extension of the Wong-Zakai theorem for stochastic evolution equations in Hilbert spaces, Stochastic Anal. Appl., 10 (1992), 471-500. doi: 10.1080/07362999208809284. Google Scholar
[52]	K. Twardowska, On the approximation theorem of the Wong-Zakai type for the functional stochastic differential equations, Probab. Math. Statist., 12 (1991), 319-334. Google Scholar
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[55]	B. Wang, Existence, stability and bifurcation of random complete and periodic solutions of stochastic parabolic equations, Nonlinear Anal., 103 (2014), 9-25. doi: 10.1016/j.na.2014.02.013. Google Scholar
[56]	B. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Discrete Contin. Dyn. Syst., 34 (2014), 269-300. doi: 10.3934/dcds.2014.34.269. Google Scholar
[57]	B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations, 253 (2012), 1544-1583. doi: 10.1016/j.jde.2012.05.015. Google Scholar
[58]	X. Wang, K. Lu and B. Wang, Wong-Zakai approximations and attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 264 (2018), 378-424. doi: 10.1016/j.jde.2017.09.006. Google Scholar
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show all references

References:

[1]	L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7. Google Scholar
[2]	V. Bally, A. Millet and M. Sanz-Solé, Approximation and support theorem in Hölder norm for parabolic stochastic partial differential equations, Ann. Probab., 23 (1995), 178-222. doi: 10.1214/aop/1176988383. Google Scholar
[3]	M. Bartuccelli, P. Constantin, C. R. Doering, J. Gibbon and M. Gisselfält, On the possibility of soft and hard turbulence in the complex Ginzburg-Landau equation, Phys. D, 44 (1990), 421-444. doi: 10.1016/0167-2789(90)90156-J. Google Scholar
[4]	Z. Brzeźniak, M. Capiński and F. Flandoli, A convergence result for stochastic partial differential equations, Stochastics, 24 (1988), 423-445. doi: 10.1080/17442508808833526. Google Scholar
[5]	Z. Brzeźniak and F. Flandoli, Almost sure approximation of Wong-Zakai type for stochastic partial differential equations, Stochastic Process. Appl., 55 (1995), 329-358. doi: 10.1016/0304-4149(94)00037-T. Google Scholar
[6]	T. Caraballo, J. A. Langa, V. S. Melnik and J. Valero, Pullback attractors for nonautonomous and stochastic multivalued dynamical systems, Set-Valued Anal., 11 (2003), 153-201. doi: 10.1023/A:1022902802385. Google Scholar
[7]	H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393. doi: 10.1007/BF01193705. Google Scholar
[8]	A. Deya, M. Jolis and L. Quer-Sardanyons, The Stratonovich heat equation: A continuity result and weak approximations, Electron. J. Probab., 18 (2013), 34 pp. doi: 10.1214/EJP.v18-2004. Google Scholar
[9]	C. R. Doering, J. D. Gibbon and C. D. Levermore, Weak and strong solutions of the complex Ginzburg-Landau equation, Phys. D, 71 (1994), 285-318. doi: 10.1016/0167-2789(94)90150-3. Google Scholar
[10]	J. Duan, P. Holmes and E. S. Titi, Global existence theory for a generalized Ginzburg-Landau equation, Nonlinearity, 5 (1992), 1303-1314. doi: 10.1088/0951-7715/5/6/005. Google Scholar
[11]	F. Flandoli, Regularity Theory and Stochastic Flow for Parabolic SPDEs, Stochastics Monographs, Vol. 9, Gordon and Breach Science Publishers, Yverdon, 1995. Google Scholar
[12]	F. Flandoli and B. Schmalfuss, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative white noise, Stochastics Stochastics Rep., 59 (1996), 21-45. doi: 10.1080/17442509608834083. Google Scholar
[13]	A. Ganguly, Wong-ZaKai type convergence in infinite dimensions, Electron. J. Probab., 18 (2013), 34 pp. doi: 10.1214/EJP.v18-2650. Google Scholar
[14]	H. Gao, M. Garrido-Atienza and B. Schmalfuss, Random attractors for stochastic evolution equations driven by fractional Brownian motion, SIAM J. Math. Anal., 46 (2014), 2281-2309. doi: 10.1137/130930662. Google Scholar
[15]	M. Garrido-Atienza, K. Lu and B. Schmalfuss, Random dynamical systems for stochastic evolution equations driven by multiplicative fractional Brownian noise with Hurst parameters $H \in (1/3, 1/2]$, SIAM J. Appl. Dyn. Syst., 15 (2016), 625-654. doi: 10.1137/15M1030303. Google Scholar
[16]	W. Grecksch and B. Schmalfuss, Approximation of the stochastic Navier-Stokes equation, Mat. Apl. Comput., 15 (1996), 227-239. Google Scholar
[17]	B. Guo and B. Wang, Finite dimensional behavior for the derivative Ginzburg-Landau equation in two spatial dimensions, Phys. D, 89 (1995), 83-99. doi: 10.1016/0167-2789(95)00216-2. Google Scholar
[18]	I. Gyöngy, On the approximation of stochastic partial differential equations I, Stochastics, 25 (1988), 59-85. doi: 10.1080/17442508808833533. Google Scholar
[19]	I. Gyöngy, On the approximation of stochastic partial differential equations II, Stochastics Stochastics Rep., 26 (1989), 129-164. doi: 10.1080/17442508908833554. Google Scholar
[20]	I. Gyöngy and A. Shmatkov, Rate of convergence of Wong-Zakai approximations for stochastic partial differential equations, Appl. Math. Optim., 54 (2006), 315-341. doi: 10.1007/s00245-006-0873-2. Google Scholar
[21]	M. Hairer and E. Pardoux, A Wong-Zakai theorem for stochastic PDEs, J. Math. Soc. Japan, 67 (2015), 1551-1604. doi: 10.2969/jmsj/06741551. Google Scholar
[22]	N. Ikeda, S. Nakao and Y. Yamato, A class of approximations of Brownian motion, Publ. Res. Inst. Math. Sci., 13 (1977/78), 285-300. doi: 10.2977/prims/1195190109. Google Scholar
[23]	N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, 2$^nd$ edition, North-Holland Publishing Co., Amsterdam, Kodansha, Ltd., Tokyo, 1989. Google Scholar
[24]	D. Kelley and I. Melbourne, Smooth approximation of stochastic differential equations, Ann. Probab., 44 (2016), 479-520. doi: 10.1214/14-AOP979. Google Scholar
[25]	F. Konecny, On Wong-Zakai approximation of stochastic differential equations, J. Multivariate Anal., 13 (1983), 605-611. doi: 10.1016/0047-259X(83)90043-X. Google Scholar
[26]	T. G. Kurtz and P. Protter, Weak limit theorems for stochastic integrals and stochastic differential equations, Ann. Probab., 19 (1991), 1035-1070. doi: 10.1214/aop/1176990334. Google Scholar
[27]	T. Kurtz and P. Protter, Wong-Zakai corrections, random evolutions, and simulation schemes for SDE, in Stochastic Analysis: Liber Amicorum for Moshe Zakai, Academic Press, Boston, (1991), 331–346. doi: 10.1016/B978-0-12-481005-1.50023-5. Google Scholar
[28]	Y. Lan and J. Shu, Dynamics of non-autonomous fractional stochastic Ginzburg-Landau equations with multiplicative noise, Commun. Pure Appl. Anal., 18 (2019), 2409-2431. doi: 10.3934/cpaa.2019109. Google Scholar
[29]	Y. Lan and J. Shu, Fractal dimension of random attractors for non-autonomous fractional stochastic Ginzburg-Landau equations with multiplicative noise, Dyn. Syst., 34 (2019), 274-300. doi: 10.1080/14689367.2018.1523368. Google Scholar
[30]	D. Li, Z. Dai and X. Liu, Long time behaviour for generalized complex Ginzburg-Landau equation, J. Math. Anal. Appl., 330 (2007), 934-948. doi: 10.1016/j.jmaa.2006.07.095. Google Scholar
[31]	D. Li and B. Guo, Asymptotic behavior of the 2D generalized stochastic Ginzburg-Landau equation with additive noise, Appl. Math. Mech. (English Ed.), 30 (2009), 945-956. doi: 10.1007/s10483-009-0801-x. Google Scholar
[32]	D. Li, K. Lu, B. Wang and X. Wang, Limiting behavior of dynamics for stochastic reaction-diffusion equations with additive noise on thin domains, Discrete Contin. Dyn. Syst., 38 (2018), 187-208. doi: 10.3934/dcds.2018009. Google Scholar
[33]	D. Li, K. Lu, B. Wang and X. Wang, Limiting dynamics for non-autonomous stochastic retarded reaction-diffusion equations on thin domains, Discrete Contin. Dyn. Syst., 39 (2019), 3717-3747. doi: 10.3934/dcds.2019151. Google Scholar
[34]	J. L. Lions, Quelques Methodes de Resolution des Problemes aux Limites Non Lineaires, Dunod, Gauthier-Villars, Paris, 1969. Google Scholar
[35]	K. Lu, and B. Wang, Wong-Zakai approximations and long term behavior of stochastic partial differential equations, J. Dynam. Differential Equations, (2019), 1341–1371. doi: 10.1007/s10884-017-9626-y. Google Scholar
[36]	K. Lu and Q. Wang, Chaotic behavior in differential equations driven by a Brownian motion, J. Differential Equations, 251 (2011), 2853-2895. doi: 10.1016/j.jde.2011.05.032. Google Scholar
[37]	E. J. McShane, Stochastic Differential Equations and Models of Random Processes. Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Vol. Ⅲ, Univ. California Press, Berkeley, Calif., 1972,263{294. Google Scholar
[38]	S. Nakao, On weak convergence of sequences of continuous local martingales, Ann. Inst. H. Poincar?Probab. Statist., 22 (1986), 371-380. Google Scholar
[39]	S. Nakao and Y. Yamato, Approximation theorem on stochastic differential equations, in Proceedings of the International Symposium on Stochastic Differential Equations, Wiley, New York-Chinchester-Brisbane, 1978,283–296. Google Scholar
[40]	A. Nowak, A Wong-Zakai type theorem for stochastic systems of Burgers equations, PanAmer. Math. J., 16 (2006), 1-25. Google Scholar
[41]	P. Protter, Approximations of solutions of stochastic differential equations driven by semimartingales, Ann. Probab., 13 (1985), 716-743. doi: 10.1214/aop/1176992905. Google Scholar
[42]	B. Schmalfuss, V. Reitmann, T. Riedrich and N. Koksch (eds.), Backward cocycles and attractors of stochastic differential equations, in International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior, Technische Universitat, Dresden, 1992,185–192. Google Scholar
[43]	J. Shen, K. Lu and W. Zhang, Heteroclinic chaotic behavior driven by a Brownian motion, J. Differential Equations, 255 (2013), 4185-4225. doi: 10.1016/j.jde.2013.08.003. Google Scholar
[44]	J. Shu, P. Li, J. Zhang and O. Liao, Random attractors for the stochastic coupled fractional Ginzburg-Landau equation with additive noise, J. Math. Phys., 56 (2015), 102702, 11 pp. doi: 10.1063/1.4934724. Google Scholar
[45]	D. W. Stroock and S. R. S. Varadhan, On the support of diffusion processes with applications to the strong maximum principle, in Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Vol. Ⅲ: Probability Theory, Univ. California Press, Berkeley, Calif., 1972,333{359. Google Scholar
[46]	H. J. Sussmann, An interpretation of stochastic differential equations as ordinary differential equations which depend on the sample point, Bull. Amer. Math. Soc., 83 (1977), 296-298. doi: 10.1090/S0002-9904-1977-14312-7. Google Scholar
[47]	H. J. Sussmann, On the gap between deterministic and stochastic ordinary differential equations, Ann. Probab., 6 (1978), 19-41. doi: 10.1214/aop/1176995608. Google Scholar
[48]	R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2$^{nd}$ edition, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3. Google Scholar
[49]	G. Tessitore and J. Zabczyk, Wong-Zakai approximations of stochastic evolution equations, J. Evol. Equ., 6 (2006), 621-655. doi: 10.1007/s00028-006-0280-9. Google Scholar
[50]	K. Twardowska, An approximation theorem of Wong-Zakai type for nonlinear stochastic partial differential equations, Stochastic Anal. Appl., 13 (1995), 601-626. doi: 10.1080/07362999508809419. Google Scholar
[51]	K. Twardowska, An extension of the Wong-Zakai theorem for stochastic evolution equations in Hilbert spaces, Stochastic Anal. Appl., 10 (1992), 471-500. doi: 10.1080/07362999208809284. Google Scholar
[52]	K. Twardowska, On the approximation theorem of the Wong-Zakai type for the functional stochastic differential equations, Probab. Math. Statist., 12 (1991), 319-334. Google Scholar
[53]	K. Twardowska, Wong-Zakai approximations for stochastic differential equations, Acta Appl. Math., 43 (1996), 317-359. doi: 10.1007/BF00047670. Google Scholar
[54]	B. Wang, Existence and upper semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms, Stoch. Dyn., 14 (2014), 1450009, 31 pp. doi: 10.1142/S0219493714500099. Google Scholar
[55]	B. Wang, Existence, stability and bifurcation of random complete and periodic solutions of stochastic parabolic equations, Nonlinear Anal., 103 (2014), 9-25. doi: 10.1016/j.na.2014.02.013. Google Scholar
[56]	B. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Discrete Contin. Dyn. Syst., 34 (2014), 269-300. doi: 10.3934/dcds.2014.34.269. Google Scholar
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Wong-Zakai approximations and asymptotic behavior of stochastic Ginzburg-Landau equations

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Wong-Zakai approximations and asymptotic behavior of stochastic Ginzburg-Landau equations

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