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January  2019, 39(1): 185-218. doi: 10.3934/dcds.2019008

Asymptotic behavior of random Navier-Stokes equations driven by Wong-Zakai approximations

1. 

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

2. 

Department of Mathematics, Brigham Young University, Provo, Utah 84602, USA

3. 

Department of Mathematics, New Mexico Institute of Mining and Technology, Socorro, NM 87801, USA

* Corresponding author

Received  October 2017 Revised  July 2018 Published  October 2018

Fund Project: The first author is supported by NSF of Chongqing grant cstc2018jcyjA0897

In this paper, we investigate the asymptotic behavior of the solutions of the two-dimensional stochastic Navier-Stokes equations via the stationary Wong-Zakai approximations given by the Wiener shift. We prove the existence and uniqueness of tempered pullback attractors for the random equations of the Wong-Zakai approximations with a Lipschitz continuous diffusion term. Under certain conditions, we also prove the convergence of solutions and random attractors of the approximate equations when the step size of approximations approaches zero.

Citation: Anhui Gu, Kening Lu, Bixiang Wang. Asymptotic behavior of random Navier-Stokes equations driven by Wong-Zakai approximations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 185-218. doi: 10.3934/dcds.2019008
References:
[1]

P. Acquistapace and B. Terreni, An approach to Itö linear equations in Hilbert spaces by approximation of white noise with coloured noise, Stochastic Anal. Appl., 2 (1984), 131-186.  doi: 10.1080/07362998408809031.  Google Scholar

[2]

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

[3]

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.  Google Scholar

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P. W. BatesH. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems, Stoch. Dyn., 6 (2006), 1-21.  doi: 10.1142/S0219493706001621.  Google Scholar

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P. W. BatesK. Lu and B. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 246 (2009), 845-869.  doi: 10.1016/j.jde.2008.05.017.  Google Scholar

[6]

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

[7]

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

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M. Capiński and N. J. Cutland, Existence of global stochastic flow and attractors for NavierStokes equations, Probab. Theory Relat. Fields, 115 (1999), 121-151.  doi: 10.1007/s004400050238.  Google Scholar

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T. CaraballoM. Garrido-AtienzaB. Schmalfuss and J. Valero, Non-autonomous and random attractors for delay random semilinear equations without uniqueness, Discrete Contin. Dyn. Syst., 21 (2008), 415-443.  doi: 10.3934/dcds.2008.21.415.  Google Scholar

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T. CaraballoJ. Real and I. Chueshov, Pullback attractors for stochastic heat equations in materials with memory, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 525-539.  doi: 10.3934/dcdsb.2008.9.525.  Google Scholar

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T. Caraballo and J. Langa, On the upper semicontinuity of cocycle attractors for non-autonomous and random dynamical systems, Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis, 10 (2003), 491-513.   Google Scholar

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T. CaraballoM. Garrido-AtienzaB. Schmalfuss and J. Valero, Asymptotic behaviour of a stochastic semilinear dissipative functional equation without uniqueness of solutions, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 439-455.  doi: 10.3934/dcdsb.2010.14.439.  Google Scholar

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T. CaraballoJ. LangaV. S. Melnik and J. Valero, Pullback attractors for nonautonomous and stochastic multivalued dynamical systems, Set-Valued Analysis, 11 (2003), 153-201.  doi: 10.1023/A:1022902802385.  Google Scholar

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I. Chueshov and M. Scheutzow, On the structure of attractors and invariant measures for a class of monotone random systems, Dynamical Systems, 19 (2004), 127-144.  doi: 10.1080/1468936042000207792.  Google Scholar

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I. Chueshov, Monotone Random Systems - Theory and Applications, Lecture Notes in Mathematics, 1779, Springer, Berlin, 2002. doi: 10.1007/b83277.  Google Scholar

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H. CrauelA. Debussche and F. Flandoli, Random attractors, J. Dyn. Diff. Equat., 9 (1997), 307-341.  doi: 10.1007/BF02219225.  Google Scholar

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

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A. DeyaM. Jolis and L. Quer-Sardanyons, The Stratonovich heat equation: A continuity result and weak approximations, Electron. J. Probab., 18 (2013), 1-34.  doi: 10.1214/EJP.v18-2004.  Google Scholar

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J. Duan and B. Schmalfuss, The 3D quasigeostrophic fluid dynamics under random forcing on boundary, Comm. Math. Sci., 1 (2003), 133-151.  doi: 10.4310/CMS.2003.v1.n1.a9.  Google Scholar

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F. Flandoli and B. Schmalfuss, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise, Stoch. Stoch. Rep., 59 (1996), 21-45.  doi: 10.1080/17442509608834083.  Google Scholar

[22]

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

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

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M. Garrido-Atienza and B. Schmalfuss, Ergodicity of the infinite dimensional fractional Brownian motion, J. Dynam. Differential Equations, 23 (2011), 671-681.  doi: 10.1007/s10884-011-9222-5.  Google Scholar

[25]

M. Garrido-AtienzaA. Ogrowsky and B. Schmalfuss, Random differential equations with random delays, Stoch. Dyn., 11 (2011), 369-388.  doi: 10.1142/S0219493711003358.  Google Scholar

[26]

M. Garrido-AtienzaB. Maslowski and B. Schmalfuss, Random attractors for stochastic equations driven by a fractional Brownian motion, International J. Bifurcation and Chaos, 20 (2010), 2761-2782.  doi: 10.1142/S0218127410027349.  Google Scholar

[27]

B. GessW. Liu and M. Rockner, Random attractors for a class of stochastic partial differential equations driven by general additive noise, J. Differential Equations, 251 (2011), 1225-1253.  doi: 10.1016/j.jde.2011.02.013.  Google Scholar

[28]

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

[29]

I. Gyöngy and A. Shmatkov, Rate of convergence of Wong-Zakai approximations for stochastic partial differential equations, Applied Mathematics and Optimization, 54 (2006), 315-341.  doi: 10.1007/s00245-006-0873-2.  Google Scholar

[30]

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

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

[32]

M. Hairer and E. Pardoux, A Wong-Zakai theorem for stochastic PDE, J. Math. Soc. Japan., 67 (2015), 1551-1604.  doi: 10.2969/jmsj/06741551.  Google Scholar

[33]

J. Huang and W. Shen, Pullback attractors for nonautonomous and random parabolic equations on non-smooth domains, Discrete Contin. Dyn. Syst., 24 (2009), 855-882.  doi: 10.3934/dcds.2009.24.855.  Google Scholar

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N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, Kodansha, Ltd., Tokyo, 1981.  Google Scholar

[35]

N. IkedaS. Nakao and Y. Yamato, A class of approximations of Brownian motion, Publ. RIMS, Kyoto Univ., 13 (1977), 285-300.  doi: 10.2977/prims/1195190109.  Google Scholar

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

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P. E. Kloeden and J. A. Langa, Flattening, squeezing and the existence of random attractors, Proc. Royal Soc. London Serie A., 463 (2007), 163-181.  doi: 10.1098/rspa.2006.1753.  Google Scholar

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T. Kurtz and P. Protter, Wong-Zakai corrections, random evolutions, and simulation schemes for sde. Stochastic Analysis: Liber Amicorum for Moshe Zakai, Academic Press, San Diego., (1991), 331-346.   Google Scholar

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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

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K. Lu and B. Wang, Wong-Zakai approximations and long term behavior of stochastic partial differential equations, J. Dyn. Diff. Equat. (2017), https://doi.org/10.1007/s10884-017-9626-y. Google Scholar

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R. Manthey, Weak convergence of solutions of the heat equation with Gaussian noise, Math. Nachr., 123 (1985), 157-168.  doi: 10.1002/mana.19851230115.  Google Scholar

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E. J. McShane, Stochastic differential equations and models of random processes, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), 3 (1972), 263-294.   Google Scholar

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S. Nakao, On weak convergence of sequences of continuous local martingale, Annales De L'I. H. P., Section B, 22 (1986), 371-380.   Google Scholar

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

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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

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Z. ShenS. Zhou and W. Shen, One-dimensional random attractor and rotation number of the stochastic damped sine-Gordon equation, J. Differential Equations, 248 (2010), 1432-1457.  doi: 10.1016/j.jde.2009.10.007.  Google Scholar

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show all references

References:
[1]

P. Acquistapace and B. Terreni, An approach to Itö linear equations in Hilbert spaces by approximation of white noise with coloured noise, Stochastic Anal. Appl., 2 (1984), 131-186.  doi: 10.1080/07362998408809031.  Google Scholar

[2]

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

[3]

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.  Google Scholar

[4]

P. W. BatesH. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems, Stoch. Dyn., 6 (2006), 1-21.  doi: 10.1142/S0219493706001621.  Google Scholar

[5]

P. W. BatesK. Lu and B. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 246 (2009), 845-869.  doi: 10.1016/j.jde.2008.05.017.  Google Scholar

[6]

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

[7]

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

[8]

M. Capiński and N. J. Cutland, Existence of global stochastic flow and attractors for NavierStokes equations, Probab. Theory Relat. Fields, 115 (1999), 121-151.  doi: 10.1007/s004400050238.  Google Scholar

[9]

T. CaraballoM. Garrido-AtienzaB. Schmalfuss and J. Valero, Non-autonomous and random attractors for delay random semilinear equations without uniqueness, Discrete Contin. Dyn. Syst., 21 (2008), 415-443.  doi: 10.3934/dcds.2008.21.415.  Google Scholar

[10]

T. CaraballoJ. Real and I. Chueshov, Pullback attractors for stochastic heat equations in materials with memory, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 525-539.  doi: 10.3934/dcdsb.2008.9.525.  Google Scholar

[11]

T. Caraballo and J. Langa, On the upper semicontinuity of cocycle attractors for non-autonomous and random dynamical systems, Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis, 10 (2003), 491-513.   Google Scholar

[12]

T. CaraballoM. Garrido-AtienzaB. Schmalfuss and J. Valero, Asymptotic behaviour of a stochastic semilinear dissipative functional equation without uniqueness of solutions, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 439-455.  doi: 10.3934/dcdsb.2010.14.439.  Google Scholar

[13]

T. CaraballoM. Garrido-Atienza and T. Taniguchi, The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion, Nonlinear Anal., 74 (2011), 3671-3684.  doi: 10.1016/j.na.2011.02.047.  Google Scholar

[14]

T. CaraballoJ. LangaV. S. Melnik and J. Valero, Pullback attractors for nonautonomous and stochastic multivalued dynamical systems, Set-Valued Analysis, 11 (2003), 153-201.  doi: 10.1023/A:1022902802385.  Google Scholar

[15]

I. Chueshov and M. Scheutzow, On the structure of attractors and invariant measures for a class of monotone random systems, Dynamical Systems, 19 (2004), 127-144.  doi: 10.1080/1468936042000207792.  Google Scholar

[16]

I. Chueshov, Monotone Random Systems - Theory and Applications, Lecture Notes in Mathematics, 1779, Springer, Berlin, 2002. doi: 10.1007/b83277.  Google Scholar

[17]

H. CrauelA. Debussche and F. Flandoli, Random attractors, J. Dyn. Diff. Equat., 9 (1997), 307-341.  doi: 10.1007/BF02219225.  Google Scholar

[18]

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

[19]

A. DeyaM. Jolis and L. Quer-Sardanyons, The Stratonovich heat equation: A continuity result and weak approximations, Electron. J. Probab., 18 (2013), 1-34.  doi: 10.1214/EJP.v18-2004.  Google Scholar

[20]

J. Duan and B. Schmalfuss, The 3D quasigeostrophic fluid dynamics under random forcing on boundary, Comm. Math. Sci., 1 (2003), 133-151.  doi: 10.4310/CMS.2003.v1.n1.a9.  Google Scholar

[21]

F. Flandoli and B. Schmalfuss, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise, Stoch. Stoch. Rep., 59 (1996), 21-45.  doi: 10.1080/17442509608834083.  Google Scholar

[22]

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

[23]

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

[24]

M. Garrido-Atienza and B. Schmalfuss, Ergodicity of the infinite dimensional fractional Brownian motion, J. Dynam. Differential Equations, 23 (2011), 671-681.  doi: 10.1007/s10884-011-9222-5.  Google Scholar

[25]

M. Garrido-AtienzaA. Ogrowsky and B. Schmalfuss, Random differential equations with random delays, Stoch. Dyn., 11 (2011), 369-388.  doi: 10.1142/S0219493711003358.  Google Scholar

[26]

M. Garrido-AtienzaB. Maslowski and B. Schmalfuss, Random attractors for stochastic equations driven by a fractional Brownian motion, International J. Bifurcation and Chaos, 20 (2010), 2761-2782.  doi: 10.1142/S0218127410027349.  Google Scholar

[27]

B. GessW. Liu and M. Rockner, Random attractors for a class of stochastic partial differential equations driven by general additive noise, J. Differential Equations, 251 (2011), 1225-1253.  doi: 10.1016/j.jde.2011.02.013.  Google Scholar

[28]

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

[29]

I. Gyöngy and A. Shmatkov, Rate of convergence of Wong-Zakai approximations for stochastic partial differential equations, Applied Mathematics and Optimization, 54 (2006), 315-341.  doi: 10.1007/s00245-006-0873-2.  Google Scholar

[30]

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

[31]

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

[32]

M. Hairer and E. Pardoux, A Wong-Zakai theorem for stochastic PDE, J. Math. Soc. Japan., 67 (2015), 1551-1604.  doi: 10.2969/jmsj/06741551.  Google Scholar

[33]

J. Huang and W. Shen, Pullback attractors for nonautonomous and random parabolic equations on non-smooth domains, Discrete Contin. Dyn. Syst., 24 (2009), 855-882.  doi: 10.3934/dcds.2009.24.855.  Google Scholar

[34]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, Kodansha, Ltd., Tokyo, 1981.  Google Scholar

[35]

N. IkedaS. Nakao and Y. Yamato, A class of approximations of Brownian motion, Publ. RIMS, Kyoto Univ., 13 (1977), 285-300.  doi: 10.2977/prims/1195190109.  Google Scholar

[36]

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

[37]

P. E. Kloeden and J. A. Langa, Flattening, squeezing and the existence of random attractors, Proc. Royal Soc. London Serie A., 463 (2007), 163-181.  doi: 10.1098/rspa.2006.1753.  Google Scholar

[38]

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

[39]

T. 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

[40]

T. Kurtz and P. Protter, Wong-Zakai corrections, random evolutions, and simulation schemes for sde. Stochastic Analysis: Liber Amicorum for Moshe Zakai, Academic Press, San Diego., (1991), 331-346.   Google Scholar

[41]

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

[42]

K. Lu and B. Wang, Wong-Zakai approximations and long term behavior of stochastic partial differential equations, J. Dyn. Diff. Equat. (2017), https://doi.org/10.1007/s10884-017-9626-y. Google Scholar

[43]

R. Manthey, Weak convergence of solutions of the heat equation with Gaussian noise, Math. Nachr., 123 (1985), 157-168.  doi: 10.1002/mana.19851230115.  Google Scholar

[44]

E. J. McShane, Stochastic differential equations and models of random processes, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), 3 (1972), 263-294.   Google Scholar

[45]

S. Nakao, On weak convergence of sequences of continuous local martingale, Annales De L'I. H. P., Section B, 22 (1986), 371-380.   Google Scholar

[46]

S. Nakao and Y. Yamato, Approximation theorem on stochastic differential equations, Proc. International Symp. S.D.E., Kyoto. 1976,283-296.  Google Scholar

[47]

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

[48]

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

[49]

B. Schmalfuss, Backward cocycles and attractors of stochastic differential equations, International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior, 185-192, Dresden, 1992. Google Scholar

[50]

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.  Google Scholar

[51]

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