doi: 10.3934/dcdsb.2021033

Approximate dynamics of a class of stochastic wave equations with white noise

1. 

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

2. 

College of Management Science, Chengdu University of Technology, Chengdu, 610059, China

* Corresponding author: Guanggan Chen

Received  August 2020 Revised  November 2020 Published  February 2021

Fund Project: The first author is supported by the National Science Foundation of China (Grants No. 11571245)

This work is concerned with a stochastic wave equation driven by a white noise. Borrowing from the invariant random cone and employing the backward solvability argument, this wave system is approximated by a finite dimensional wave equation with a white noise. Especially, the finite dimension is explicit, accurate and determined by the coefficient of this wave system; and further originating from an Ornstein-Uhlenbek process and applying Banach space norm estimation, this wave system is approximated by a finite dimensional wave equation with a smooth colored noise.

Citation: Guanggan Chen, Qin Li, Yunyun Wei. Approximate dynamics of a class of stochastic wave equations with white noise. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021033
References:
[1]

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

[2]

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

[3]

S. Cerrai and M. I. Freidlin, On the Smoluchowski-Kramers approximation for a system with an infinite number of degrees of freedom, Probab. Theory Rel., 135 (2006), 363-394.  doi: 10.1007/s00440-005-0465-0.  Google Scholar

[4]

G. ChenJ. Duan and J. Zhang, Approximating dynamics of a singularly perturbed stochastic wave equation with a random dynamical boundary condition, SIAM. J. Math. Anal., 45 (2013), 2790-2814.  doi: 10.1137/12088968X.  Google Scholar

[5]

P. L. Chow, Asymptotics of solutions to semilinear stochastic wave equations, Ann. Appl. Probab., 16 (2006), 757-780.  doi: 10.1214/105051606000000141.  Google Scholar

[6]

P. L. Chow, Stochastic wave equations with polynomial nonlinearity, Ann. Appl. Probab., 12 (2002), 361-381.  doi: 10.1214/aoap/1015961168.  Google Scholar

[7] G. da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, UK, 1992.  doi: 10.1017/CBO9780511666223.  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]

J. DuanK. Lu and B. Schmalfuß, Smooth stable and unstable manifolds for stochastic evolutionary equations, J. Dyn. Differ. Equ., 16 (2004), 949-972.  doi: 10.1007/s10884-004-7830-z.  Google Scholar

[10]

X. Fan and Y. Wang, Fractal dimension of attractors for a stochastic wave equation with nonlinear damping and white noise, Stoch. Anal. Appl., 25 (2007), 381-396.  doi: 10.1080/07362990601139602.  Google Scholar

[11]

J. Garcia-Ojalvo and J. M. Sancho, Noise in Spatially Extended Systems, Springer-Verlag, Berlin, 1999. doi: 10.1007/978-1-4612-1536-3.  Google Scholar

[12]

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

[13]

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

[14]

J. Hale and G. Raugel, Upper semicontinuity of the attractor for a singularly perturbed hyperbolic equation, J. Differ. Equations, 73 (1988), 197-214.  doi: 10.1016/0022-0396(88)90104-0.  Google Scholar

[15]

W. Horsthemke and R. Lefever, Noise-induced Transitions: Theory and Applications in Physics, Chemistry, and Biology, Springer Series in Synergetics, Berlin, Springer, 1984. doi: 10.1007/3-540-36852-3.  Google Scholar

[16]

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

[17]

T. Jiang, X. Liu and J. Duan, A Wong-Zakai approximation for random invariant manifolds, J. Math. Phys., 58 (2017), 122701. doi: 10.1063/1.5017932.  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]

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

[20]

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

[21]

K. Lu and B. Schmalfuß, Invariant manifolds for stochastic wave equations, J. Differ. Equations, 236 (2007), 460-492.  doi: 10.1016/j.jde.2006.09.024.  Google Scholar

[22]

Y. Lv and W. Wang, Limiting dynamics for stochastic wave equations, J. Differ. Equations, 244 (2008), 1-23.  doi: 10.1016/j.jde.2007.10.009.  Google Scholar

[23]

Y. Lv, W. Wang and A. J. Roberts, Approximation of the random inertial manifold of singularly perturbed stochastic wave equations, Stoch. Dynam., 14 (2014), 1350018. doi: 10.1142/S0219493713500184.  Google Scholar

[24]

X. Mora, Finite-dimensional attracting invariant manifolds for damped semilinear wave equations, Contributions to Nonlinear Partial Differential Equations, 2 (1985), 172-183.   Google Scholar

[25]

C. Mueller, Long time existence for the wave equation with a noise term, Ann. Probab., 25 (1997), 133-151.  doi: 10.1214/aop/1024404282.  Google Scholar

[26]

S. Nakao, On weak convergence of sequences of continuous local martingale, Ann. I. H. Poincare B, 22 (1986), 371-380.   Google Scholar

[27]

E. Pardoux and A. Piatnitski, Homogenization of a singular random one-dimensional PDE with time-varying coefficients, Ann. Probab., 40 (2012), 1316-1356.  doi: 10.1214/11-AOP650.  Google Scholar

[28]

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

[29] M. Reed and B. Simon, Methods of Modern Mathematical Physics Ⅱ, Academic Press, New York, 1975.   Google Scholar
[30]

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

[31]

G. Tessitore and J. Zabczyk, Wong-Zakai approximation of stochastic evolution equations, J. Evol. Equ., 6 (2006), 621-655.  doi: 10.1007/s00028-006-0280-9.  Google Scholar

[32]

X. WangK. Lu and B. Wang, Wong-Zakai approximations and attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differ. Equations, 264 (2018), 378-424.  doi: 10.1016/j.jde.2017.09.006.  Google Scholar

[33]

W. Wang and J. Duan, A dynamical approximation for stochastic partial differential equations, J. Math. Phys., 48 (2007), 102701. doi: 10.1063/1.2800164.  Google Scholar

[34]

G. Whitham, Linear and Nonlinear Waves, Wiley, New York, 1974.  Google Scholar

[35]

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

[36]

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

[37]

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

show all references

References:
[1]

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

[2]

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

[3]

S. Cerrai and M. I. Freidlin, On the Smoluchowski-Kramers approximation for a system with an infinite number of degrees of freedom, Probab. Theory Rel., 135 (2006), 363-394.  doi: 10.1007/s00440-005-0465-0.  Google Scholar

[4]

G. ChenJ. Duan and J. Zhang, Approximating dynamics of a singularly perturbed stochastic wave equation with a random dynamical boundary condition, SIAM. J. Math. Anal., 45 (2013), 2790-2814.  doi: 10.1137/12088968X.  Google Scholar

[5]

P. L. Chow, Asymptotics of solutions to semilinear stochastic wave equations, Ann. Appl. Probab., 16 (2006), 757-780.  doi: 10.1214/105051606000000141.  Google Scholar

[6]

P. L. Chow, Stochastic wave equations with polynomial nonlinearity, Ann. Appl. Probab., 12 (2002), 361-381.  doi: 10.1214/aoap/1015961168.  Google Scholar

[7] G. da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, UK, 1992.  doi: 10.1017/CBO9780511666223.  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]

J. DuanK. Lu and B. Schmalfuß, Smooth stable and unstable manifolds for stochastic evolutionary equations, J. Dyn. Differ. Equ., 16 (2004), 949-972.  doi: 10.1007/s10884-004-7830-z.  Google Scholar

[10]

X. Fan and Y. Wang, Fractal dimension of attractors for a stochastic wave equation with nonlinear damping and white noise, Stoch. Anal. Appl., 25 (2007), 381-396.  doi: 10.1080/07362990601139602.  Google Scholar

[11]

J. Garcia-Ojalvo and J. M. Sancho, Noise in Spatially Extended Systems, Springer-Verlag, Berlin, 1999. doi: 10.1007/978-1-4612-1536-3.  Google Scholar

[12]

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

[13]

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

[14]

J. Hale and G. Raugel, Upper semicontinuity of the attractor for a singularly perturbed hyperbolic equation, J. Differ. Equations, 73 (1988), 197-214.  doi: 10.1016/0022-0396(88)90104-0.  Google Scholar

[15]

W. Horsthemke and R. Lefever, Noise-induced Transitions: Theory and Applications in Physics, Chemistry, and Biology, Springer Series in Synergetics, Berlin, Springer, 1984. doi: 10.1007/3-540-36852-3.  Google Scholar

[16]

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

[17]

T. Jiang, X. Liu and J. Duan, A Wong-Zakai approximation for random invariant manifolds, J. Math. Phys., 58 (2017), 122701. doi: 10.1063/1.5017932.  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]

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

[20]

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

[21]

K. Lu and B. Schmalfuß, Invariant manifolds for stochastic wave equations, J. Differ. Equations, 236 (2007), 460-492.  doi: 10.1016/j.jde.2006.09.024.  Google Scholar

[22]

Y. Lv and W. Wang, Limiting dynamics for stochastic wave equations, J. Differ. Equations, 244 (2008), 1-23.  doi: 10.1016/j.jde.2007.10.009.  Google Scholar

[23]

Y. Lv, W. Wang and A. J. Roberts, Approximation of the random inertial manifold of singularly perturbed stochastic wave equations, Stoch. Dynam., 14 (2014), 1350018. doi: 10.1142/S0219493713500184.  Google Scholar

[24]

X. Mora, Finite-dimensional attracting invariant manifolds for damped semilinear wave equations, Contributions to Nonlinear Partial Differential Equations, 2 (1985), 172-183.   Google Scholar

[25]

C. Mueller, Long time existence for the wave equation with a noise term, Ann. Probab., 25 (1997), 133-151.  doi: 10.1214/aop/1024404282.  Google Scholar

[26]

S. Nakao, On weak convergence of sequences of continuous local martingale, Ann. I. H. Poincare B, 22 (1986), 371-380.   Google Scholar

[27]

E. Pardoux and A. Piatnitski, Homogenization of a singular random one-dimensional PDE with time-varying coefficients, Ann. Probab., 40 (2012), 1316-1356.  doi: 10.1214/11-AOP650.  Google Scholar

[28]

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

[29] M. Reed and B. Simon, Methods of Modern Mathematical Physics Ⅱ, Academic Press, New York, 1975.   Google Scholar
[30]

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

[31]

G. Tessitore and J. Zabczyk, Wong-Zakai approximation of stochastic evolution equations, J. Evol. Equ., 6 (2006), 621-655.  doi: 10.1007/s00028-006-0280-9.  Google Scholar

[32]

X. WangK. Lu and B. Wang, Wong-Zakai approximations and attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differ. Equations, 264 (2018), 378-424.  doi: 10.1016/j.jde.2017.09.006.  Google Scholar

[33]

W. Wang and J. Duan, A dynamical approximation for stochastic partial differential equations, J. Math. Phys., 48 (2007), 102701. doi: 10.1063/1.2800164.  Google Scholar

[34]

G. Whitham, Linear and Nonlinear Waves, Wiley, New York, 1974.  Google Scholar

[35]

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

[36]

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

[37]

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

[1]

Jun Shen, Kening Lu, Bixiang Wang. Invariant manifolds and foliations for random differential equations driven by colored noise. Discrete & Continuous Dynamical Systems, 2020, 40 (11) : 6201-6246. doi: 10.3934/dcds.2020276

[2]

Renhai Wang, Yangrong Li, Bixiang Wang. Random dynamics of fractional nonclassical diffusion equations driven by colored noise. Discrete & Continuous Dynamical Systems, 2019, 39 (7) : 4091-4126. doi: 10.3934/dcds.2019165

[3]

Bixiang Wang. Random attractors for non-autonomous stochastic wave equations with multiplicative noise. Discrete & Continuous Dynamical Systems, 2014, 34 (1) : 269-300. doi: 10.3934/dcds.2014.34.269

[4]

Anhui Gu, Boling Guo, Bixiang Wang. Long term behavior of random Navier-Stokes equations driven by colored noise. Discrete & Continuous Dynamical Systems - B, 2020, 25 (7) : 2495-2532. doi: 10.3934/dcdsb.2020020

[5]

Renhai Wang, Bixiang Wang. Random dynamics of lattice wave equations driven by infinite-dimensional nonlinear noise. Discrete & Continuous Dynamical Systems - B, 2020, 25 (7) : 2461-2493. doi: 10.3934/dcdsb.2020019

[6]

Lin Shi, Xuemin Wang, Dingshi Li. Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5367-5386. doi: 10.3934/cpaa.2020242

[7]

Yanfeng Guo, Jinqiao Duan, Donglong Li. Approximation of random invariant manifolds for a stochastic Swift-Hohenberg equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1701-1715. doi: 10.3934/dcdss.2016071

[8]

Anhui Gu, Bixiang Wang. Asymptotic behavior of random fitzhugh-nagumo systems driven by colored noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1689-1720. doi: 10.3934/dcdsb.2018072

[9]

Giuseppe Da Prato. An integral inequality for the invariant measure of some finite dimensional stochastic differential equation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3015-3027. doi: 10.3934/dcdsb.2016085

[10]

Xiaojun Li, Xiliang Li, Kening Lu. Random attractors for stochastic parabolic equations with additive noise in weighted spaces. Communications on Pure & Applied Analysis, 2018, 17 (3) : 729-749. doi: 10.3934/cpaa.2018038

[11]

Chi Phan. Random attractor for stochastic Hindmarsh-Rose equations with multiplicative noise. Discrete & Continuous Dynamical Systems - B, 2020, 25 (8) : 3233-3256. doi: 10.3934/dcdsb.2020060

[12]

Xingni Tan, Fuqi Yin, Guihong Fan. Random exponential attractor for stochastic discrete long wave-short wave resonance equation with multiplicative white noise. Discrete & Continuous Dynamical Systems - B, 2020, 25 (8) : 3153-3170. doi: 10.3934/dcdsb.2020055

[13]

Jun Shen, Kening Lu, Bixiang Wang. Convergence and center manifolds for differential equations driven by colored noise. Discrete & Continuous Dynamical Systems, 2019, 39 (8) : 4797-4840. doi: 10.3934/dcds.2019196

[14]

Bixiang Wang. Mean-square random invariant manifolds for stochastic differential equations. Discrete & Continuous Dynamical Systems, 2021, 41 (3) : 1449-1468. doi: 10.3934/dcds.2020324

[15]

T. Tachim Medjo. The exponential behavior of the stochastic primitive equations in two dimensional space with multiplicative noise. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 177-197. doi: 10.3934/dcdsb.2010.14.177

[16]

Shengfan Zhou, Min Zhao. Fractal dimension of random attractor for stochastic non-autonomous damped wave equation with linear multiplicative white noise. Discrete & Continuous Dynamical Systems, 2016, 36 (5) : 2887-2914. doi: 10.3934/dcds.2016.36.2887

[17]

Zhaojuan Wang, Shengfan Zhou. Existence and upper semicontinuity of random attractors for non-autonomous stochastic strongly damped wave equation with multiplicative noise. Discrete & Continuous Dynamical Systems, 2017, 37 (5) : 2787-2812. doi: 10.3934/dcds.2017120

[18]

Lingyu Li, Zhang Chen. Asymptotic behavior of non-autonomous random Ginzburg-Landau equation driven by colored noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3303-3333. doi: 10.3934/dcdsb.2020233

[19]

Zhaojuan Wang, Shengfan Zhou. Random attractor and random exponential attractor for stochastic non-autonomous damped cubic wave equation with linear multiplicative white noise. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4767-4817. doi: 10.3934/dcds.2018210

[20]

Renhai Wang, Yangrong Li. Backward compactness and periodicity of random attractors for stochastic wave equations with varying coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4145-4167. doi: 10.3934/dcdsb.2019054

2019 Impact Factor: 1.27

Article outline

[Back to Top]