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doi: 10.3934/dcdsb.2022054
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## Limiting behavior of invariant measures of highly nonlinear stochastic retarded lattice systems

 School of Mathematics, Southwest Jiaotong University, Chengdu, Sichuan 610031, China

* Corresponding author

Received  September 2021 Revised  January 2022 Early access March 2022

Fund Project: This work was supported by NSFC (11971394 and 11871049), Sichuan Science and Technology Program (2019YJ0215) and Fundamental Research Funds for the Central Universities (2682021ZTPY057)

This paper deals with the limiting behavior of invariant measures of the highly nonlinear stochastic retarded lattice systems. Although invariant measures of stochastic retarded lattice system has been studied recently, there is so far no result of invariant measure of stochastic retarded lattice systems with highly nonlinear drift or diffusion terms. We first show the existence of invariant measures of the systems. We then prove that any limit point of a tight sequence of invariant measures of the stochastic retarded lattice systems must be an invariant measure of the corresponding limiting system as the intensity of noise converges or the time-delay approaches zero.

Citation: Yusen Lin, Dingshi Li. Limiting behavior of invariant measures of highly nonlinear stochastic retarded lattice systems. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022054
##### References:
 [1] P. W. Bates, H. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems, Stoch. Dyn., 6 (2006), 1-21.  doi: 10.1142/S0219493706001621. [2] P. W. Bates, K. 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. [3] H. Bessaih, M. Garrido-Atienza, X. Han and B. Schmalfuss, Stochastic lattice dynamical systems with fractional noise, SIAM J. Math. Anal., 49 (2017), 1495-1518.  doi: 10.1137/16M1085504. [4] T. Caraballo, X. Han, B. 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. [5] 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. [6] T. Caraballo, F. Morillas and J. Valero, Random attractors for stochastic lattice systems with non-Lipschitz nonlinearity, J. Difference Equ. Appl., 17 (2011), 161-184.  doi: 10.1080/10236198.2010.549010. [7] T. Caraballo, F. Morillas and J. Valero, Attractors of stochastic lattice dynamical systems with a multiplicative noise and non-Lipschitz nonlinearity, J. Differential Equations, 253 (2012), 667-693.  doi: 10.1016/j.jde.2012.03.020. [8] Z. Chen, X. Li and B. Wang, Invariant measures of stochastic delay lattice systems, Discrete Contin., Dyn. Syst. Ser. B, 26 (2021), 3235–3269. doi: 10.3934/dcdsb.2020226. [9] X. Han, W. 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. [10] D. Li, B. Wang and X. Wang, Periodic measures of stochastic delay lattice systems, J. Differential Equations, 272 (2021), 74-104.  doi: 10.1016/j.jde.2020.09.034. [11] D. Li, B. Wang and X. Wang, Limiting behavior of invariant measures of stochastic delay lattice systems, J. Dynam. Differential Equations, 2021. doi: 10.1007/s10884-021-10011-7. [12] X. Mao, Stochastic Differential Equations and Applications, Second edition. Horwood Publishing Limited, Chichester, 2008. doi: 10.1533/9780857099402. [13] M. Sui, Y. Wang, X. Han and P. E. Kloeden, Random recurrent neural networks with delays, J. Differential Equations, 269 (2020), 8597-8639.  doi: 10.1016/j.jde.2020.06.008. [14] B. Wang, Dynamics of stochastic reaction-diffusion lattice systems driven by nonlinear noise, J. Math. Anal. Appl., 477 (2019), 104-132.  doi: 10.1016/j.jmaa.2019.04.015. [15] X. Wang, S. Li and D. Xu, Random attractors for second-order stochastic lattice dynamical systems, Nonlinear Anal., 72 (2010), 483-494.  doi: 10.1016/j.na.2009.06.094. [16] X. Wang, K. Lu and B. Wang, Exponential stability of non-autonomous stochastic delay lattice systems driven by a multiplicative white noise, J. Dynam. Differential Equations, 28 (2016), 1309-1335.  doi: 10.1007/s10884-015-9448-8. [17] R. Wang and B. Wang, Random dynamics of $p$-Laplacian lattice systems driven by infinite-dimensional nonlinear noise, Stochastic Process. Appl., 130 (2020), 7431-7462.  doi: 10.1016/j.spa.2020.08.002. [18] R. Wang and B. Wang, Asymptotic behavior of stochastic Schr$\ddot{o}$dinger lattice systems driven by nonlinear noise, Stoch. Anal. Appl., 38 (2020), 213-237.  doi: 10.1080/07362994.2019.1679646. [19] Z. Wang and S. Zhou, Existence and upper semicontinuity of attractors for non-autonomous stochastic lattice systems with random coupled coefficients, Commun. Pure Appl. Anal., 15 (2016), 2221-2245.  doi: 10.3934/cpaa.2016035. [20] W. Yan, Y. Li and S. Ji, Random attractors for first order stochastic retarded lattice dynamical systems, J. Math. Phys., 51 (2010), 032702, 17pp. doi: 10.1063/1.3319566. [21] C. Zhang and L. Zhao, The attractors for 2nd-order stochastic delay lattice systems, Discrete Contin. Dyn. Syst., 37 (2017), 575-590.  doi: 10.3934/dcds.2017023.

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##### References:
 [1] P. W. Bates, H. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems, Stoch. Dyn., 6 (2006), 1-21.  doi: 10.1142/S0219493706001621. [2] P. W. Bates, K. 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. [3] H. Bessaih, M. Garrido-Atienza, X. Han and B. Schmalfuss, Stochastic lattice dynamical systems with fractional noise, SIAM J. Math. Anal., 49 (2017), 1495-1518.  doi: 10.1137/16M1085504. [4] T. Caraballo, X. Han, B. 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. [5] 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. [6] T. Caraballo, F. Morillas and J. Valero, Random attractors for stochastic lattice systems with non-Lipschitz nonlinearity, J. Difference Equ. Appl., 17 (2011), 161-184.  doi: 10.1080/10236198.2010.549010. [7] T. Caraballo, F. Morillas and J. Valero, Attractors of stochastic lattice dynamical systems with a multiplicative noise and non-Lipschitz nonlinearity, J. Differential Equations, 253 (2012), 667-693.  doi: 10.1016/j.jde.2012.03.020. [8] Z. Chen, X. Li and B. Wang, Invariant measures of stochastic delay lattice systems, Discrete Contin., Dyn. Syst. Ser. B, 26 (2021), 3235–3269. doi: 10.3934/dcdsb.2020226. [9] X. Han, W. 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. [10] D. Li, B. Wang and X. Wang, Periodic measures of stochastic delay lattice systems, J. Differential Equations, 272 (2021), 74-104.  doi: 10.1016/j.jde.2020.09.034. [11] D. Li, B. Wang and X. Wang, Limiting behavior of invariant measures of stochastic delay lattice systems, J. Dynam. Differential Equations, 2021. doi: 10.1007/s10884-021-10011-7. [12] X. Mao, Stochastic Differential Equations and Applications, Second edition. Horwood Publishing Limited, Chichester, 2008. doi: 10.1533/9780857099402. [13] M. Sui, Y. Wang, X. Han and P. E. Kloeden, Random recurrent neural networks with delays, J. Differential Equations, 269 (2020), 8597-8639.  doi: 10.1016/j.jde.2020.06.008. [14] B. Wang, Dynamics of stochastic reaction-diffusion lattice systems driven by nonlinear noise, J. Math. Anal. Appl., 477 (2019), 104-132.  doi: 10.1016/j.jmaa.2019.04.015. [15] X. Wang, S. Li and D. Xu, Random attractors for second-order stochastic lattice dynamical systems, Nonlinear Anal., 72 (2010), 483-494.  doi: 10.1016/j.na.2009.06.094. [16] X. Wang, K. Lu and B. Wang, Exponential stability of non-autonomous stochastic delay lattice systems driven by a multiplicative white noise, J. Dynam. Differential Equations, 28 (2016), 1309-1335.  doi: 10.1007/s10884-015-9448-8. [17] R. Wang and B. Wang, Random dynamics of $p$-Laplacian lattice systems driven by infinite-dimensional nonlinear noise, Stochastic Process. Appl., 130 (2020), 7431-7462.  doi: 10.1016/j.spa.2020.08.002. [18] R. Wang and B. Wang, Asymptotic behavior of stochastic Schr$\ddot{o}$dinger lattice systems driven by nonlinear noise, Stoch. Anal. Appl., 38 (2020), 213-237.  doi: 10.1080/07362994.2019.1679646. [19] Z. Wang and S. Zhou, Existence and upper semicontinuity of attractors for non-autonomous stochastic lattice systems with random coupled coefficients, Commun. Pure Appl. Anal., 15 (2016), 2221-2245.  doi: 10.3934/cpaa.2016035. [20] W. Yan, Y. Li and S. Ji, Random attractors for first order stochastic retarded lattice dynamical systems, J. Math. Phys., 51 (2010), 032702, 17pp. doi: 10.1063/1.3319566. [21] C. Zhang and L. Zhao, The attractors for 2nd-order stochastic delay lattice systems, Discrete Contin. Dyn. Syst., 37 (2017), 575-590.  doi: 10.3934/dcds.2017023.
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