October  2019, 24(10): 5737-5767. doi: 10.3934/dcdsb.2019104

Asymptotic behavior of random lattice dynamical systems and their Wong-Zakai approximations

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

* Corresponding author: Anhui Gu

Received  June 2018 Revised  October 2018 Published  October 2019 Early access  June 2019

Fund Project: This work is supported by NSF of Chongqing grant cstc2018jcyjA0897.

In this paper, we study the Wong-Zakai approximations given by a smoothed approximation of the white noise and their associated long term pathwise behavior for the stochastic lattice dynamical systems. To be exactly, we first establish the existence of the random attractor for the random lattice dynamical system driven by the smoothed noise and then show the convergence of solutions and random attractors to these of stochastic lattice dynamical systems driven by a multiplicative noise and an additive white noise, respectively, when the perturbation parameters tend to zero.

Citation: Anhui Gu. Asymptotic behavior of random lattice dynamical systems and their Wong-Zakai approximations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (10) : 5737-5767. doi: 10.3934/dcdsb.2019104
References:
[1]

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

[2]

V. BallyA. Millet and M. Sanz-Sole, Approximation and support theorem in Hölder norm for parabolic stochastic partial differential equations, Ann. Probab., 23 (1995), 178-222.  doi: 10.1214/aop/1176988383.

[3]

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

[4]

P. W. BatesK. Lu and B. Wang, Attractors for lattice dynamical systems, Internat. J. Bifur. Chaos, 11 (2001), 143-153.  doi: 10.1142/S0218127401002031.

[5]

P. W. BatesK. Lu and B. Wang, Attractors of non-autonomous stochastic lattice systems in weighted spaces, Physica D, 289 (2014), 32-50.  doi: 10.1016/j.physd.2014.08.004.

[6]

H. BessaihM. Garrido-AtienzaX. Han and B. Schmalfuss, Stochastic lattice dynamical systems with fractional noise, SIAM J. Math. Anal., 49 (2017), 1495-1518.  doi: 10.1137/16M1085504.

[7]

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.

[8]

T. CaraballoX. HanB. Schmalfuss and J. Valero, Random attractors for stochastic lattice dynamical systems with infinite multiplicative white noise, Nonlinear Analysis, 130 (2016), 255-278.  doi: 10.1016/j.na.2015.09.025.

[9]

T. Caraballo and K. Lu, Attractors for stochastic lattice dynamical systems with a multiplicative noise., Front. Math. China, 3 (2008), 317-335.  doi: 10.1007/s11464-008-0028-7.

[10]

A. Carvalho, J. Langa and J. Robinson, Attractors for Infinite-Dimensional Nonautonomous Dynamical Systems, Springer, New York, 2013. doi: 10.1007/978-1-4614-4581-4.

[11]

V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Colloquium Publications, American Mathematical Society, Providence, 2002.

[12]

S. N. Chow, Lattice dynamical systems, Dynamical Systems, 1–102, Lecture Notes in Math., 1822, Springer, Berlin, 2003. doi: 10.1007/978-3-540-45204-1_1.

[13]

I. Chueshov, Monotone Random Dynamical Systems Theory and Applications, Springer-Verlag, Berlin, 2002. doi: 10.1007/b83277.

[14]

X. Fan and H. Chen, Attractors for the stochastic reaction-diffusion equation driven by linear multiplicative noise with a variable coefficient, J. Math. Anal. Appl., 398 (2013), 715-728.  doi: 10.1016/j.jmaa.2012.09.027.

[15]

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

[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.

[17]

X. HanW. Shen and S. Zhou, Random attractors for stochastic lattice dynamical systems in weighted spaces, J. Differential Equations, 250 (2011), 1235-1266.  doi: 10.1016/j.jde.2010.10.018.

[18]

P. Kloeden and M. Rasmussen, Non-autonomous Dynamical Systems, American Mathematical Society, Providence, 2011. doi: 10.1090/surv/176.

[19]

K. Lu and B. Wang, Wong-Zakai approximations and long term behavior of stochastic partial differential equations, J. Dynam. Differential Equations, (2017), 1–31. doi: 10.1007/s10884-017-9626-y.

[20]

A. Pazy, Semigroups of Linear Operator and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[21]

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.

[22]

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.

[23]

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.

[24]

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.

[25]

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.

[26]

C. ZhaoG. Xue and G. Łukaszewicz, Pullback attractors and invariant measures for discrete Klein-Gordon-Schrödinger equations, Discrete Cont. Dyn. Syst. B, 23 (2018), 4021-4044.  doi: 10.3934/dcdsb.2018122.

[27]

S. Zhou, Random exponential attractor for cocycle and application to non-autonomous stochastic lattice systems with multiplicative white noise, J. Differential Equations, 263 (2017), 2247-2279.  doi: 10.1016/j.jde.2017.03.044.

[28]

S. Zhou and X. Han, Pullback exponential attractors for non-autonomous lattice systems, J. Dynam. Differential Equations, 24 (2012), 601-631.  doi: 10.1007/s10884-012-9260-7.

show all references

References:
[1]

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

[2]

V. BallyA. Millet and M. Sanz-Sole, Approximation and support theorem in Hölder norm for parabolic stochastic partial differential equations, Ann. Probab., 23 (1995), 178-222.  doi: 10.1214/aop/1176988383.

[3]

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

[4]

P. W. BatesK. Lu and B. Wang, Attractors for lattice dynamical systems, Internat. J. Bifur. Chaos, 11 (2001), 143-153.  doi: 10.1142/S0218127401002031.

[5]

P. W. BatesK. Lu and B. Wang, Attractors of non-autonomous stochastic lattice systems in weighted spaces, Physica D, 289 (2014), 32-50.  doi: 10.1016/j.physd.2014.08.004.

[6]

H. BessaihM. Garrido-AtienzaX. Han and B. Schmalfuss, Stochastic lattice dynamical systems with fractional noise, SIAM J. Math. Anal., 49 (2017), 1495-1518.  doi: 10.1137/16M1085504.

[7]

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.

[8]

T. CaraballoX. HanB. Schmalfuss and J. Valero, Random attractors for stochastic lattice dynamical systems with infinite multiplicative white noise, Nonlinear Analysis, 130 (2016), 255-278.  doi: 10.1016/j.na.2015.09.025.

[9]

T. Caraballo and K. Lu, Attractors for stochastic lattice dynamical systems with a multiplicative noise., Front. Math. China, 3 (2008), 317-335.  doi: 10.1007/s11464-008-0028-7.

[10]

A. Carvalho, J. Langa and J. Robinson, Attractors for Infinite-Dimensional Nonautonomous Dynamical Systems, Springer, New York, 2013. doi: 10.1007/978-1-4614-4581-4.

[11]

V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Colloquium Publications, American Mathematical Society, Providence, 2002.

[12]

S. N. Chow, Lattice dynamical systems, Dynamical Systems, 1–102, Lecture Notes in Math., 1822, Springer, Berlin, 2003. doi: 10.1007/978-3-540-45204-1_1.

[13]

I. Chueshov, Monotone Random Dynamical Systems Theory and Applications, Springer-Verlag, Berlin, 2002. doi: 10.1007/b83277.

[14]

X. Fan and H. Chen, Attractors for the stochastic reaction-diffusion equation driven by linear multiplicative noise with a variable coefficient, J. Math. Anal. Appl., 398 (2013), 715-728.  doi: 10.1016/j.jmaa.2012.09.027.

[15]

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

[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.

[17]

X. HanW. Shen and S. Zhou, Random attractors for stochastic lattice dynamical systems in weighted spaces, J. Differential Equations, 250 (2011), 1235-1266.  doi: 10.1016/j.jde.2010.10.018.

[18]

P. Kloeden and M. Rasmussen, Non-autonomous Dynamical Systems, American Mathematical Society, Providence, 2011. doi: 10.1090/surv/176.

[19]

K. Lu and B. Wang, Wong-Zakai approximations and long term behavior of stochastic partial differential equations, J. Dynam. Differential Equations, (2017), 1–31. doi: 10.1007/s10884-017-9626-y.

[20]

A. Pazy, Semigroups of Linear Operator and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[21]

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.

[22]

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.

[23]

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.

[24]

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.

[25]

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.

[26]

C. ZhaoG. Xue and G. Łukaszewicz, Pullback attractors and invariant measures for discrete Klein-Gordon-Schrödinger equations, Discrete Cont. Dyn. Syst. B, 23 (2018), 4021-4044.  doi: 10.3934/dcdsb.2018122.

[27]

S. Zhou, Random exponential attractor for cocycle and application to non-autonomous stochastic lattice systems with multiplicative white noise, J. Differential Equations, 263 (2017), 2247-2279.  doi: 10.1016/j.jde.2017.03.044.

[28]

S. Zhou and X. Han, Pullback exponential attractors for non-autonomous lattice systems, J. Dynam. Differential Equations, 24 (2012), 601-631.  doi: 10.1007/s10884-012-9260-7.

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