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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 and Continuous Dynamical Systems, 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.

[2]

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

[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.

[4]

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

[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.

[6]

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

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[16]

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

[17]

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

[18]

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

[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.

[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.

[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.

[22]

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

[23]

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

[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.

[25]

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

[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.

[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.

[28]

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

[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.

[30]

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

[31]

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

[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.

[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.

[34]

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

[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.

[36]

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

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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. 

[45]

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

[46]

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

[47]

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

[48]

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

[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.

[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.

[51]

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.

[52]

D. W. Stook and S. R. S. Varadhan, On the support of diffusion processes with applications to the strong maximum principle, Proc. 6-th Berkeley Symp. on Math. Stat. and Prob., 3 (1972), 333-359. 

[53]

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.

[54]

H. J. Sussmann, On the gap between deterministic and stochastic ordinary differential equations, Ann. Probab., 6 (1978), 19-41.  doi: 10.1214/aop/1176995608.

[55]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.

[56]

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.

[57]

B. Wang, Asymptotic behavior of stochastic wave equations with critical exponents on $\mathbb{R}^3$, Trans. Amer. Math. Soc., 363 (2011), 3639-3663.  doi: 10.1090/S0002-9947-2011-05247-5.

[58]

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.

[59]

B. Wang, Existence and upper semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms, Stoch. Dyn., 14 (2014), 1450009, 31pp. doi: 10.1142/S0219493714500099.

[60]

B. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Discrete Contin. Dyn. Syst. Ser. A, 34 (2014), 269-300.  doi: 10.3934/dcds.2014.34.269.

[61]

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.

[62]

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

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.

[2]

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

[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.

[4]

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

[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.

[6]

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

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[16]

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

[17]

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

[18]

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

[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.

[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.

[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.

[22]

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

[23]

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

[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.

[25]

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

[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.

[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.

[28]

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

[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.

[30]

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

[31]

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

[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.

[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.

[34]

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

[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.

[36]

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

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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. 

[45]

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

[46]

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

[47]

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

[48]

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

[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.

[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.

[51]

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.

[52]

D. W. Stook and S. R. S. Varadhan, On the support of diffusion processes with applications to the strong maximum principle, Proc. 6-th Berkeley Symp. on Math. Stat. and Prob., 3 (1972), 333-359. 

[53]

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.

[54]

H. J. Sussmann, On the gap between deterministic and stochastic ordinary differential equations, Ann. Probab., 6 (1978), 19-41.  doi: 10.1214/aop/1176995608.

[55]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.

[56]

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.

[57]

B. Wang, Asymptotic behavior of stochastic wave equations with critical exponents on $\mathbb{R}^3$, Trans. Amer. Math. Soc., 363 (2011), 3639-3663.  doi: 10.1090/S0002-9947-2011-05247-5.

[58]

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.

[59]

B. Wang, Existence and upper semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms, Stoch. Dyn., 14 (2014), 1450009, 31pp. doi: 10.1142/S0219493714500099.

[60]

B. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Discrete Contin. Dyn. Syst. Ser. A, 34 (2014), 269-300.  doi: 10.3934/dcds.2014.34.269.

[61]

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.

[62]

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

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