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

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

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  November 2020 Early access  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.

Citation: 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
References:
[1]

L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.

[2]

V. BallyA. 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.

[3]

M. BartuccelliP. ConstantinC. R. DoeringJ. 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.

[4]

Z. BrzeźniakM. Capiński and F. Flandoli, A convergence result for stochastic partial differential equations, Stochastics, 24 (1988), 423-445.  doi: 10.1080/17442508808833526.

[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.

[6]

T. CaraballoJ. A. LangaV. 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.

[7]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705.

[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.

[9]

C. R. DoeringJ. 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.

[10]

J. DuanP. 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.

[11]

F. Flandoli, Regularity Theory and Stochastic Flow for Parabolic SPDEs, Stochastics Monographs, Vol. 9, Gordon and Breach Science Publishers, Yverdon, 1995.

[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.

[13]

A. Ganguly, Wong-ZaKai type convergence in infinite dimensions, Electron. J. Probab., 18 (2013), 34 pp. doi: 10.1214/EJP.v18-2650.

[14]

H. GaoM. 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.

[15]

M. Garrido-AtienzaK. 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.

[16]

W. Grecksch and B. Schmalfuss, Approximation of the stochastic Navier-Stokes equation, Mat. Apl. Comput., 15 (1996), 227-239. 

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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.

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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.

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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.

[22]

N. IkedaS. 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.

[23]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, 2$^nd$ edition, North-Holland Publishing Co., Amsterdam, Kodansha, Ltd., Tokyo, 1989.

[24]

D. Kelley and I. Melbourne, Smooth approximation of stochastic differential equations, Ann. Probab., 44 (2016), 479-520.  doi: 10.1214/14-AOP979.

[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.

[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.

[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.

[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.

[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.

[30]

D. LiZ. 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.

[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.

[32]

D. LiK. LuB. 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.

[33]

D. LiK. LuB. 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.

[34]

J. L. Lions, Quelques Methodes de Resolution des Problemes aux Limites Non Lineaires, Dunod, Gauthier-Villars, Paris, 1969.

[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.

[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.

[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.

[38]

S. Nakao, On weak convergence of sequences of continuous local martingales, Ann. Inst. H. Poincar?Probab. Statist., 22 (1986), 371-380. 

[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.

[40]

A. Nowak, A Wong-Zakai type theorem for stochastic systems of Burgers equations, PanAmer. Math. J., 16 (2006), 1-25. 

[41]

P. Protter, Approximations of solutions of stochastic differential equations driven by semimartingales, Ann. Probab., 13 (1985), 716-743.  doi: 10.1214/aop/1176992905.

[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.

[43]

J. ShenK. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[53]

K. Twardowska, Wong-Zakai approximations for stochastic differential equations, Acta Appl. Math., 43 (1996), 317-359.  doi: 10.1007/BF00047670.

[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.

[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.

[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.

[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.

[58]

X. WangK. 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.

[59]

E. Wong and M. Zakai, On the convergence of ordinary integrals to stochastic integrals, Ann. Math. Statist., 36 (1965), 1560-1564.  doi: 10.1214/aoms/1177699916.

[60]

E. Wong and M. Zakai, On the relation between ordinary and stochastic diferetnial equations, Internat. J. Engrg. Sci., 3 (1965), 213-229.  doi: 10.1016/0020-7225(65)90045-5.

show all references

References:
[1]

L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.

[2]

V. BallyA. 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.

[3]

M. BartuccelliP. ConstantinC. R. DoeringJ. 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.

[4]

Z. BrzeźniakM. Capiński and F. Flandoli, A convergence result for stochastic partial differential equations, Stochastics, 24 (1988), 423-445.  doi: 10.1080/17442508808833526.

[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.

[6]

T. CaraballoJ. A. LangaV. 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.

[7]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705.

[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.

[9]

C. R. DoeringJ. 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.

[10]

J. DuanP. 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.

[11]

F. Flandoli, Regularity Theory and Stochastic Flow for Parabolic SPDEs, Stochastics Monographs, Vol. 9, Gordon and Breach Science Publishers, Yverdon, 1995.

[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.

[13]

A. Ganguly, Wong-ZaKai type convergence in infinite dimensions, Electron. J. Probab., 18 (2013), 34 pp. doi: 10.1214/EJP.v18-2650.

[14]

H. GaoM. 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.

[15]

M. Garrido-AtienzaK. 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.

[16]

W. Grecksch and B. Schmalfuss, Approximation of the stochastic Navier-Stokes equation, Mat. Apl. Comput., 15 (1996), 227-239. 

[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.

[18]

I. Gyöngy, On the approximation of stochastic partial differential equations I, Stochastics, 25 (1988), 59-85.  doi: 10.1080/17442508808833533.

[19]

I. Gyöngy, On the approximation of stochastic partial differential equations II, Stochastics Stochastics Rep., 26 (1989), 129-164.  doi: 10.1080/17442508908833554.

[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.

[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.

[22]

N. IkedaS. 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.

[23]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, 2$^nd$ edition, North-Holland Publishing Co., Amsterdam, Kodansha, Ltd., Tokyo, 1989.

[24]

D. Kelley and I. Melbourne, Smooth approximation of stochastic differential equations, Ann. Probab., 44 (2016), 479-520.  doi: 10.1214/14-AOP979.

[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.

[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.

[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.

[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.

[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.

[30]

D. LiZ. 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.

[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.

[32]

D. LiK. LuB. 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.

[33]

D. LiK. LuB. 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.

[34]

J. L. Lions, Quelques Methodes de Resolution des Problemes aux Limites Non Lineaires, Dunod, Gauthier-Villars, Paris, 1969.

[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.

[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.

[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.

[38]

S. Nakao, On weak convergence of sequences of continuous local martingales, Ann. Inst. H. Poincar?Probab. Statist., 22 (1986), 371-380. 

[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.

[40]

A. Nowak, A Wong-Zakai type theorem for stochastic systems of Burgers equations, PanAmer. Math. J., 16 (2006), 1-25. 

[41]

P. Protter, Approximations of solutions of stochastic differential equations driven by semimartingales, Ann. Probab., 13 (1985), 716-743.  doi: 10.1214/aop/1176992905.

[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.

[43]

J. ShenK. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[53]

K. Twardowska, Wong-Zakai approximations for stochastic differential equations, Acta Appl. Math., 43 (1996), 317-359.  doi: 10.1007/BF00047670.

[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.

[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.

[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.

[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.

[58]

X. WangK. 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.

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