doi: 10.3934/dcdsb.2020207

Wong-Zakai approximations and attractors for stochastic wave equations driven by additive noise

1. 

Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China

2. 

School of Mathematics, Southwest Jiaotong University, Chengdu, Sichuan 610031, China

* Corresponding author: Jun Shen, junshen85@163.com

Received  October 2019 Published  July 2020

Fund Project: This work was supported by NSFC (11831012, 11971394 and 11871049), Fundamental Research Funds for the Central Universities (YJ201646, 2682019LK02), Sichuan Science and Technology Program (2019YJ0215) and International Visiting Program for Excellent Young Scholars of SCU

In this paper, we study the Wong-Zakai approximations given by a stationary process via Euler approximation of Brownian motion and the associated long term behavior of the stochastic wave equation driven by an additive white noise on unbounded domains. We first prove the existence and uniqueness of tempered pullback attractors for stochastic wave equation and its Wong-Zakai approximation. Then, we show that the attractor of the Wong-Zakai approximate equation converges to the one of the stochastic wave equation driven by additive noise as the correlation time of noise approaches zero.

Citation: Xiaohu Wang, Dingshi Li, Jun Shen. Wong-Zakai approximations and attractors for stochastic wave equations driven by additive noise. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020207
References:
[1]

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

[2]

J. Ball, Global attractors for damped semilinear wave equations, Discrete Contin. Dynam. Syst., 10 (2004), 31-52.  doi: 10.3934/dcds.2004.10.31.  Google Scholar

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J. DuanK. Lu and B. Schmalfuss, Invariant manifolds for stochastic partial differential equations, Ann. Probab., 31 (2003), 2109-2135.  doi: 10.1214/aop/1068646380.  Google Scholar

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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.  Google Scholar

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Z. GuoX. YanW. Wang and X. Liu, Approximate the dynamical behavior for stochastic systems by Wong-Zakai approaching, J. Math. Anal. Appl., 457 (2018), 214-232.  doi: 10.1016/j.jmaa.2017.08.004.  Google Scholar

[14]

A. GuK. Lu and B. Wang, Asymptotic behavior of random Navier-Stokes equations driven by Wong-Zakai approximations, Discrete Contin. Dynam. Syst., 39 (2019), 185-218.  doi: 10.3934/dcds.2019008.  Google Scholar

[15]

A. Gu and B. Wang, Asymptotic behavior of random Fitzhugh-Nagumo systems driven by colored noise, Discrete Contin. Dynam. Syst. Ser. B, 23 (2018), 1689-1720.  doi: 10.3934/dcdsb.2018072.  Google Scholar

[16]

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

[17]

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H. LiY. You and J. Tu, Random attractors and averaging for non-autonomous stochastic wave equations with nonlinear damping, J. Differential Equations, 258 (2015), 148-190.  doi: 10.1016/j.jde.2014.09.007.  Google Scholar

[20]

K. Lu and B. Wang, Wong-Zakai approximations and long term behavior of stochastic partial differential equations, J. Dynam. Differential Equations, 31 (2019), 1341-1371.  doi: 10.1007/s10884-017-9626-y.  Google Scholar

[21]

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

[22]

J. Shen and K. Lu, Wong-Zakai approximations and center manifolds of stochastic differential equations, J. Differential Equations, 263 (2017), 4929-4977.  doi: 10.1016/j.jde.2017.06.005.  Google Scholar

[23]

J. ShenK. Lu and B. Wang, Convergence and center manifolds for differential equations driven by colored noise, Discrete Contin. Dynam. Syst., 39 (2019), 4797-4840.  doi: 10.3934/dcds.2019196.  Google Scholar

[24]

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

[25]

B. Schmalfuss, Backward cocycle and attractors of stochastic differential equations, in International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior (eds. V. Reitmann, T. Riedrich and N. Koksch), Technische Universität, Dresden, (1992), 185–192. Google Scholar

[26]

D. W. Stroock and S. R. S. Varadhan, On the support of diffusion processes with applications to the strong maximum principle, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, 3 (1972), 333-359.   Google Scholar

[27]

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[28]

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

[29]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4684-0313-8.  Google Scholar

[30]

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.  Google Scholar

[31]

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

[32]

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

[33]

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

[34]

B. Wang, Multivalued non-autonomous random dynamical systems for wave equations without uniqueness, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2011-2051.  doi: 10.3934/dcdsb.2017119.  Google Scholar

[35]

B. Wang and X. Gao, Random attractors for wave equations on unbounded domains, Discrete Contin. Dyn. Syst., (2009), 800–809. doi: 10.3934/proc.2009.2009.800.  Google Scholar

[36]

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.  Google Scholar

[37]

Z. Wang and S. Zhou, Random attractor for stochastic non-autonomous damped wave equation with critical exponent, Discrete Contin. Dyn. Syst., 37 (2017), 545-573.  doi: 10.3934/dcds.2017022.  Google Scholar

[38]

Z. Wang and S. Zhou, Random attractor and random exponential attractor for stochastic non-autonomous damped cubic wave equation with linear multiplicative white noise, Discrete Contin. Dyn. Syst., 38 (2018), 4767-4817.  doi: 10.3934/dcds.2018210.  Google Scholar

[39]

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

[40]

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

[41]

X. YanX. Liu and M. Yang, Random attractors of stochastic partial differential equations: A smooth approximation approach, Stoch. Anal. Appl., 35 (2017), 1007-1029.  doi: 10.1080/07362994.2017.1345317.  Google Scholar

[42]

S. ZhouF. Yin and Z. Ou, Random attractor for damped nonlinear wave equations with white noise, SIAM J. Appl. Dyn. Syst., 4 (2005), 883-903.  doi: 10.1137/050623097.  Google Scholar

show all references

References:
[1]

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

[2]

J. Ball, Global attractors for damped semilinear wave equations, Discrete Contin. Dynam. Syst., 10 (2004), 31-52.  doi: 10.3934/dcds.2004.10.31.  Google Scholar

[3]

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

[4]

W.J. BeynB. GessP. Lescot and M. Röckner, The global random attractor for a class of stochastic porous media equations, Comm. Partial Differential Equations, 36 (2011), 446-469.  doi: 10.1080/03605302.2010.523919.  Google Scholar

[5]

Z. Brzezniak 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]

H. CrauelA. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341.  doi: 10.1007/BF02219225.  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]

J. DuanK. Lu and B. Schmalfuss, Invariant manifolds for stochastic partial differential equations, Ann. Probab., 31 (2003), 2109-2135.  doi: 10.1214/aop/1068646380.  Google Scholar

[9]

F. Flandoli, Regularity Theory and Stochastic Flows for Parabolic SPDEs, Stochastics Monographs 9, Gordon and Breach, London, 1995.  Google Scholar

[10]

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

[11]

H. Fu, X. Liu, J. Liu and X. Wang, On smooth approximation for random attractor of stochastic partial differential equations with multiplicative noise, Stoch. Dyn., 18 (2018), 1850040, 22pp. doi: 10.1142/s0219493718500405.  Google Scholar

[12]

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.  Google Scholar

[13]

Z. GuoX. YanW. Wang and X. Liu, Approximate the dynamical behavior for stochastic systems by Wong-Zakai approaching, J. Math. Anal. Appl., 457 (2018), 214-232.  doi: 10.1016/j.jmaa.2017.08.004.  Google Scholar

[14]

A. GuK. Lu and B. Wang, Asymptotic behavior of random Navier-Stokes equations driven by Wong-Zakai approximations, Discrete Contin. Dynam. Syst., 39 (2019), 185-218.  doi: 10.3934/dcds.2019008.  Google Scholar

[15]

A. Gu and B. Wang, Asymptotic behavior of random Fitzhugh-Nagumo systems driven by colored noise, Discrete Contin. Dynam. Syst. Ser. B, 23 (2018), 1689-1720.  doi: 10.3934/dcdsb.2018072.  Google Scholar

[16]

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

[17]

T. JiangX. Liu and J. Duan, Approximation for random stable manifolds under multiplicative correlated noise, Discrete Contin. Dynam. Syst. Ser. B, 21 (2016), 3163-3174.  doi: 10.3934/dcdsb.2016091.  Google Scholar

[18]

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

[19]

H. LiY. You and J. Tu, Random attractors and averaging for non-autonomous stochastic wave equations with nonlinear damping, J. Differential Equations, 258 (2015), 148-190.  doi: 10.1016/j.jde.2014.09.007.  Google Scholar

[20]

K. Lu and B. Wang, Wong-Zakai approximations and long term behavior of stochastic partial differential equations, J. Dynam. Differential Equations, 31 (2019), 1341-1371.  doi: 10.1007/s10884-017-9626-y.  Google Scholar

[21]

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

[22]

J. Shen and K. Lu, Wong-Zakai approximations and center manifolds of stochastic differential equations, J. Differential Equations, 263 (2017), 4929-4977.  doi: 10.1016/j.jde.2017.06.005.  Google Scholar

[23]

J. ShenK. Lu and B. Wang, Convergence and center manifolds for differential equations driven by colored noise, Discrete Contin. Dynam. Syst., 39 (2019), 4797-4840.  doi: 10.3934/dcds.2019196.  Google Scholar

[24]

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

[25]

B. Schmalfuss, Backward cocycle and attractors of stochastic differential equations, in International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior (eds. V. Reitmann, T. Riedrich and N. Koksch), Technische Universität, Dresden, (1992), 185–192. Google Scholar

[26]

D. W. Stroock and S. R. S. Varadhan, On the support of diffusion processes with applications to the strong maximum principle, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, 3 (1972), 333-359.   Google Scholar

[27]

H. J. Sussmann, An interpretation of stochastic differential equations as ordinary differential equations which depend on the sample point, Bull. Am. Math. Soc., 83 (1977), 296-298.  doi: 10.1090/S0002-9904-1977-14312-7.  Google Scholar

[28]

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

[29]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4684-0313-8.  Google Scholar

[30]

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.  Google Scholar

[31]

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

[32]

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

[33]

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

[34]

B. Wang, Multivalued non-autonomous random dynamical systems for wave equations without uniqueness, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2011-2051.  doi: 10.3934/dcdsb.2017119.  Google Scholar

[35]

B. Wang and X. Gao, Random attractors for wave equations on unbounded domains, Discrete Contin. Dyn. Syst., (2009), 800–809. doi: 10.3934/proc.2009.2009.800.  Google Scholar

[36]

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.  Google Scholar

[37]

Z. Wang and S. Zhou, Random attractor for stochastic non-autonomous damped wave equation with critical exponent, Discrete Contin. Dyn. Syst., 37 (2017), 545-573.  doi: 10.3934/dcds.2017022.  Google Scholar

[38]

Z. Wang and S. Zhou, Random attractor and random exponential attractor for stochastic non-autonomous damped cubic wave equation with linear multiplicative white noise, Discrete Contin. Dyn. Syst., 38 (2018), 4767-4817.  doi: 10.3934/dcds.2018210.  Google Scholar

[39]

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

[40]

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

[41]

X. YanX. Liu and M. Yang, Random attractors of stochastic partial differential equations: A smooth approximation approach, Stoch. Anal. Appl., 35 (2017), 1007-1029.  doi: 10.1080/07362994.2017.1345317.  Google Scholar

[42]

S. ZhouF. Yin and Z. Ou, Random attractor for damped nonlinear wave equations with white noise, SIAM J. Appl. Dyn. Syst., 4 (2005), 883-903.  doi: 10.1137/050623097.  Google Scholar

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