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Asymptotic behaviour of the solutions for a weakly damped anisotropic sixth-order Schrödinger type equation in $ \mathbb{R}^2 $
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 |
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.
References:
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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. |
[2] |
L. Arnold, Random Dynamical Systems, Springer-Verlag, Heidelberg, 1998.
doi: 10.1007/978-3-662-12878-7. |
[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. |
[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. |
[5] |
P. L. Chow,
Asymptotics of solutions to semilinear stochastic wave equations, Ann. Appl. Probab., 16 (2006), 757-780.
doi: 10.1214/105051606000000141. |
[6] |
P. L. Chow,
Stochastic wave equations with polynomial nonlinearity, Ann. Appl. Probab., 12 (2002), 361-381.
doi: 10.1214/aoap/1015961168. |
[7] |
G. da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, UK, 1992.
doi: 10.1017/CBO9780511666223.![]() ![]() ![]() |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[18] |
D. Kelley and I. Melbourne,
Smooth approximation of stochastic differential equations, Ann. Probab., 44 (2016), 479-520.
doi: 10.1214/14-AOP979. |
[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. |
[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. |
[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. |
[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. |
[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. |
[24] |
X. Mora,
Finite-dimensional attracting invariant manifolds for damped semilinear wave equations, Contributions to Nonlinear Partial Differential Equations, 2 (1985), 172-183.
|
[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. |
[26] |
S. Nakao,
On weak convergence of sequences of continuous local martingale, Ann. I. H. Poincare B, 22 (1986), 371-380.
|
[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. |
[28] |
P. Protter,
Approximations of solutions of stochastic differential equations driven by semimartingales, Ann. Probab., 13 (1985), 716-743.
doi: 10.1214/aop/1176992905. |
[29] |
M. Reed and B. Simon, Methods of Modern Mathematical Physics Ⅱ, Academic Press, New York, 1975.
![]() ![]() |
[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. |
[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. |
[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. |
[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. |
[34] |
G. Whitham, Linear and Nonlinear Waves, Wiley, New York, 1974. |
[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. |
[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. |
[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. |
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. |
[2] |
L. Arnold, Random Dynamical Systems, Springer-Verlag, Heidelberg, 1998.
doi: 10.1007/978-3-662-12878-7. |
[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. |
[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. |
[5] |
P. L. Chow,
Asymptotics of solutions to semilinear stochastic wave equations, Ann. Appl. Probab., 16 (2006), 757-780.
doi: 10.1214/105051606000000141. |
[6] |
P. L. Chow,
Stochastic wave equations with polynomial nonlinearity, Ann. Appl. Probab., 12 (2002), 361-381.
doi: 10.1214/aoap/1015961168. |
[7] |
G. da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, UK, 1992.
doi: 10.1017/CBO9780511666223.![]() ![]() ![]() |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[18] |
D. Kelley and I. Melbourne,
Smooth approximation of stochastic differential equations, Ann. Probab., 44 (2016), 479-520.
doi: 10.1214/14-AOP979. |
[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. |
[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. |
[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. |
[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. |
[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. |
[24] |
X. Mora,
Finite-dimensional attracting invariant manifolds for damped semilinear wave equations, Contributions to Nonlinear Partial Differential Equations, 2 (1985), 172-183.
|
[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. |
[26] |
S. Nakao,
On weak convergence of sequences of continuous local martingale, Ann. I. H. Poincare B, 22 (1986), 371-380.
|
[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. |
[28] |
P. Protter,
Approximations of solutions of stochastic differential equations driven by semimartingales, Ann. Probab., 13 (1985), 716-743.
doi: 10.1214/aop/1176992905. |
[29] |
M. Reed and B. Simon, Methods of Modern Mathematical Physics Ⅱ, Academic Press, New York, 1975.
![]() ![]() |
[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. |
[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. |
[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. |
[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. |
[34] |
G. Whitham, Linear and Nonlinear Waves, Wiley, New York, 1974. |
[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. |
[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. |
[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. |
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