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January  2022, 27(1): 73-101. doi: 10.3934/dcdsb.2021033

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

Guanggan Chen 1,, , Qin Li 1, and  Yunyun Wei 2,

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  January 2022 Early access  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.

Keywords: Stochastic wave equations, finite dimensional approximation, invariant random cone, colored noise.
Mathematics Subject Classification: Primary: 37L55, 35L05, 37D10; Secondary: 58J65.
Citation: Guanggan Chen, Qin Li, Yunyun Wei. Approximate dynamics of a class of stochastic wave equations with white noise. Discrete & Continuous Dynamical Systems - B, 2022, 27 (1) : 73-101. 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

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J. Duan, K. 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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J. Duan, K. 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

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Z. Guo, X. Yan, W. 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

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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

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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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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

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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. Wang, K. 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. Yan, X. 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. Chen, J. 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. Duan, K. 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. Duan, K. 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. Guo, X. Yan, W. 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. Ikeda, S. 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. Wang, K. 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. Yan, X. 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

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