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  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 & 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.  Google Scholar

[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.  Google Scholar

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

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

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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.  Google Scholar

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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.  Google Scholar

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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

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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

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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.  Google Scholar

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

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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.  Google Scholar

[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.  Google Scholar

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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]

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.  Google Scholar

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

[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.  Google Scholar

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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.  Google Scholar

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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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

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

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[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.  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]

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.  Google Scholar

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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