# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2022123
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## Periodic solutions in distribution of stochastic lattice differential equations

 1 School of Mathematics and Statistics & Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun 130024, China 2 School of Mathematics, Jilin University, Changchun 130012, China

* Corresponding author: Xiaomeng Jiang

Received  January 2022 Revised  May 2022 Early access June 2022

Fund Project: The second author was supposed by NSFC grant 12071175, National Basic Research Program of China grant 2013CB834100, Special Funds of Provincial Industrial Innovation of Jilin Province China grant 2017C028-1, Project of Science and Technology Development of Jilin Province China grant 20190201302JC. The third author was supported by NSFC grant 11901231, 12071175, Special Funds of Provincial Industrial Innovation of Jilin Province China grant 2017C028-1, Project of Science and Technology Development of Jilin Province China grant 20190201302JC and Natural Science Foundation of Jilin Province grant 20200201253JC

In this paper, we consider stochastic lattice differential equations (SLDEs). Firstly, we discuss the well-posedness of solutions for SLDEs. Then, via upper and lower solutions, we obtain a pair of monotone sequences starting at them respectively, and we prove the existence of periodic solutions in distribution.

Citation: Xinping Zhou, Yong Li, Xiaomeng Jiang. Periodic solutions in distribution of stochastic lattice differential equations. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022123
##### References:
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Valero, Random attractors for stochastic lattice dynamical systems with infinite multiplicative white noise, Nonlinear Anal., 130 (2016), 255-278.  doi: 10.1016/j.na.2015.09.025. [7] T. Caraballo, F. Morillas and J. Valero, Attractors of stochastic lattice dynamical systems with a multiplicative noise and non-Lipschitz nonlinearities, J. Differential Equations, 253 (2012), 667-693.  doi: 10.1016/j.jde.2012.03.020. [8] D. Cheban and Z. Liu, Periodic, quasi-periodic, almost periodic, almost automorphic, Birkhoff recurrent and Poisson stable solutions for stochastic differential equations, J. Differential Equations, 269 (2020), 3652-3685.  doi: 10.1016/j.jde.2020.03.014. [9] F. Chen, Y. Han, Y. Li and X. Yang, Periodic solutions of Fokker-Planck equations, J. Differential Equations, 263 (2017), 285-298.  doi: 10.1016/j.jde.2017.02.032. [10] S.-N. Chow, J. Mallet-Paret and W. Shen, Traveling waves in lattice dynamical systems, J. Differential Equations, 149 (1998), 248-291.  doi: 10.1006/jdeq.1998.3478. [11] L. O. Chua and L. Yang, Cellular neural networks: Theory, IEEE Trans. Circuits and Systems, 35 (1988), 1257-1272.  doi: 10.1109/31.7600. [12] L. O. Chua and L. Yang, Cellular neural networks: Applications, IEEE Trans. Circuits and Systems, 35 (1988), 1273-1290.  doi: 10.1109/31.7601. [13] G. Da Prato and C. Tudor, Periodic and almost periodic solutions for semilinear stochastic equations, Stochastic Anal. Appl., 13 (1995), 13-33.  doi: 10.1080/07362999508809380. [14] T. Erneux and G. Nicolis, Propagating waves in discrete bistable reaction-diffusion systems, Phys. D, 67 (1993), 237-244.  doi: 10.1016/0167-2789(93)90208-I. [15] C. Feng, B. Qu and H. Zhao, Random quasi-periodic paths and quasi-periodic measures of stochastic differential equations, J. Differential Equations, 286 (2021), 119-163.  doi: 10.1016/j.jde.2021.03.022. [16] C. Feng, H. Zhao and B. Zhou, Pathwise random periodic solutions of stochastic differential equations, J. Differential Equations, 251 (2011), 119-149.  doi: 10.1016/j.jde.2011.03.019. [17] X. Han and P. E. Kloeden, Non-autonomous lattice systems with switching effects and delayed recovery, J. Differential Equations, 261 (2016), 2986-3009.  doi: 10.1016/j.jde.2016.05.015. [18] X. Han, P. E. Kloeden and S. Sonner, Discretisation of global attractors for lattice dynamical systems, J. Dynam. Differential Equations, 32 (2020), 1457-1474.  doi: 10.1007/s10884-019-09770-1. [19] 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. [20] C. Ji, X. Yang and Y. Li, Periodic solutions for SDEs through upper and lower solutions, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 4737-4754.  doi: 10.3934/dcdsb.2020122. [21] M. Ji, W. Qi, Z. Shen and Y. Yi, Existence of periodic probability solutions to Fokker-Planck equations with applications, J. Funct. Anal., 277 (2019), 108281, 41 pp. doi: 10.1016/j.jfa.2019.108281. [22] M. Ji, W. Qi, Z. Shen and Y. Yi, Convergence to periodic probability solutions in Fokker-Planck equations, SIAM J. Math. Anal., 53 (2021), 1958-1992.  doi: 10.1137/20M1319127. [23] X. Jiang and Y. Li, Wong-Zakai approximations and periodic solutions in distribution of dissipative stochastic differential equations, J. Differential Equations, 274 (2021), 652-765.  doi: 10.1016/j.jde.2020.10.022. [24] X. Jiang, Y. Li and X. Yang, Existence of periodic solutions in distribution for stochastic Newtonian systems, J. Stat. Phys., 181 (2020), 329-363.  doi: 10.1007/s10955-020-02583-3. [25] R. Kapral, Discrete models for chemically reacting systems, J. Math. Chem., 6 (1991), 113-163.  doi: 10.1007/BF01192578. [26] 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. [27] Y. Li, Z. Liu and W. Wang, Almost periodic solutions and stable solutions for stochastic differential equations, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 5927-5944.  doi: 10.3934/dcdsb.2019113. [28] Z. Liu and K. Sun, Almost automorphic solutions for stochastic differential equations driven by Lévy noise, J. Funct. Anal., 266 (2014), 1115-1149.  doi: 10.1016/j.jfa.2013.11.011. [29] Z. Liu and W. Wang, Favard separation method for almost periodic stochastic differential equations, J. Differential Equations, 260 (2016), 8109-8136.  doi: 10.1016/j.jde.2016.02.019. [30] Y. Lv and J. Sun, Dynamical behavior for stochastic lattice systems, Chaos Solitons Fractals, 27 (2006), 1080-1090.  doi: 10.1016/j.chaos.2005.04.089. [31] S. Ma and X. Zou, Propagation and its failure in a lattice delayed differential equation with global interaction, J. Differential Equations, 212 (2005), 129-190.  doi: 10.1016/j.jde.2004.07.014. [32] X. Ma, X.-B. Shu and J. Mao, Existence of almost periodic solutions for fractional impulsive neutral stochastic differential equations with infinite delay, Stoch. Dyn., 20 (2020), 2050003, 31 pp. doi: 10.1142/S0219493720500033. [33] X. Mao, Stochastic Differential Equations and Applications, 2$^{nd}$ edition, Horwood Publishing Limited, Chichester, 2008. doi: 10.1533/9780857099402. [34] J. Shen, J. Zhao, K. Lu and B. Wang, The Wong-Zakai approximations of invariant manifolds and foliations for stochastic evolution equations, J. Differential Equations, 266 (2019), 4568-4623.  doi: 10.1016/j.jde.2018.10.008. [35] W. Shen, Traveling waves in time periodic lattice differential equations, Nonlinear Anal., 54 (2003), 319-339.  doi: 10.1016/S0362-546X(03)00065-8. [36] B. Wang, Dynamics of systems on infinite lattices, J. Differential Equations, 221 (2006), 224-245.  doi: 10.1016/j.jde.2005.01.003. [37] X. Wang, J. Shen, K. Lu and B. Wang, Wong-Zakai approximations and random attractors for non-autonomous stochastic lattice systems, J. Differential Equations, 280 (2021), 477-516.  doi: 10.1016/j.jde.2021.01.026. [38] Y. Wang, J. Xu and P. E. Kloeden, Asymptotic behavior of stochastic lattice systems with a Caputo fractional time derivative, Nonlinear Anal., 135 (2016), 205-222.  doi: 10.1016/j.na.2016.01.020. [39] W. Yan, Y. Li and S. Ji, Random attractors for first order stochastic retarded lattice dynamical systems, J. Math. Phys., 51 (2010), 032702, 17 pp. doi: 10.1063/1.3319566. [40] C. Zhao and S. Zhou, Attractors of retarded first order lattice systems, Nonlinearity, 20 (2007), 1987-2006.  doi: 10.1088/0951-7715/20/8/010. [41] H. Zhao and Z.-H. Zheng, Random periodic solutions of random dynamical systems, J. Differential Equations, 246 (2009), 2020-2038.  doi: 10.1016/j.jde.2008.10.011. [42] 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.

show all references

##### References:
 [1] P. W. Bates, X. Chen and A. J. J. Chmaj, Traveling waves of bistable dynamics on a lattice, SIAM J. Math. Anal., 35 (2003), 520-546.  doi: 10.1137/S0036141000374002. [2] P. W. Bates, H. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems, Stoch. Dyn., 6 (2006), 1-21.  doi: 10.1142/S0219493706001621. [3] P. W. Bates, K. Lu and B. Wang, Attractors of non-autonomous stochastic lattice systems in weighted spaces, Phys. D, 289 (2014), 32-50.  doi: 10.1016/j.physd.2014.08.004. [4] J. Bell and C. Cosner, Threshold behavior and propagation for nonlinear differential-difference systems motivated by modeling myelinated axons, Quart. Appl. Math., 42 (1984), 1-14.  doi: 10.1090/qam/736501. [5] H. Bessaih, M. J. 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. [6] T. Caraballo, X. Han, B. Schmalfuss and J. Valero, Random attractors for stochastic lattice dynamical systems with infinite multiplicative white noise, Nonlinear Anal., 130 (2016), 255-278.  doi: 10.1016/j.na.2015.09.025. [7] T. Caraballo, F. Morillas and J. Valero, Attractors of stochastic lattice dynamical systems with a multiplicative noise and non-Lipschitz nonlinearities, J. Differential Equations, 253 (2012), 667-693.  doi: 10.1016/j.jde.2012.03.020. [8] D. Cheban and Z. Liu, Periodic, quasi-periodic, almost periodic, almost automorphic, Birkhoff recurrent and Poisson stable solutions for stochastic differential equations, J. Differential Equations, 269 (2020), 3652-3685.  doi: 10.1016/j.jde.2020.03.014. [9] F. Chen, Y. Han, Y. Li and X. Yang, Periodic solutions of Fokker-Planck equations, J. Differential Equations, 263 (2017), 285-298.  doi: 10.1016/j.jde.2017.02.032. [10] S.-N. Chow, J. Mallet-Paret and W. Shen, Traveling waves in lattice dynamical systems, J. Differential Equations, 149 (1998), 248-291.  doi: 10.1006/jdeq.1998.3478. [11] L. O. Chua and L. Yang, Cellular neural networks: Theory, IEEE Trans. Circuits and Systems, 35 (1988), 1257-1272.  doi: 10.1109/31.7600. [12] L. O. Chua and L. Yang, Cellular neural networks: Applications, IEEE Trans. Circuits and Systems, 35 (1988), 1273-1290.  doi: 10.1109/31.7601. [13] G. Da Prato and C. Tudor, Periodic and almost periodic solutions for semilinear stochastic equations, Stochastic Anal. Appl., 13 (1995), 13-33.  doi: 10.1080/07362999508809380. [14] T. Erneux and G. Nicolis, Propagating waves in discrete bistable reaction-diffusion systems, Phys. D, 67 (1993), 237-244.  doi: 10.1016/0167-2789(93)90208-I. [15] C. Feng, B. Qu and H. Zhao, Random quasi-periodic paths and quasi-periodic measures of stochastic differential equations, J. Differential Equations, 286 (2021), 119-163.  doi: 10.1016/j.jde.2021.03.022. [16] C. Feng, H. Zhao and B. Zhou, Pathwise random periodic solutions of stochastic differential equations, J. Differential Equations, 251 (2011), 119-149.  doi: 10.1016/j.jde.2011.03.019. [17] X. Han and P. E. Kloeden, Non-autonomous lattice systems with switching effects and delayed recovery, J. Differential Equations, 261 (2016), 2986-3009.  doi: 10.1016/j.jde.2016.05.015. [18] X. Han, P. E. Kloeden and S. Sonner, Discretisation of global attractors for lattice dynamical systems, J. Dynam. Differential Equations, 32 (2020), 1457-1474.  doi: 10.1007/s10884-019-09770-1. [19] 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. [20] C. Ji, X. Yang and Y. Li, Periodic solutions for SDEs through upper and lower solutions, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 4737-4754.  doi: 10.3934/dcdsb.2020122. [21] M. Ji, W. Qi, Z. Shen and Y. Yi, Existence of periodic probability solutions to Fokker-Planck equations with applications, J. Funct. Anal., 277 (2019), 108281, 41 pp. doi: 10.1016/j.jfa.2019.108281. [22] M. Ji, W. Qi, Z. Shen and Y. Yi, Convergence to periodic probability solutions in Fokker-Planck equations, SIAM J. Math. Anal., 53 (2021), 1958-1992.  doi: 10.1137/20M1319127. [23] X. Jiang and Y. Li, Wong-Zakai approximations and periodic solutions in distribution of dissipative stochastic differential equations, J. Differential Equations, 274 (2021), 652-765.  doi: 10.1016/j.jde.2020.10.022. [24] X. Jiang, Y. Li and X. Yang, Existence of periodic solutions in distribution for stochastic Newtonian systems, J. Stat. Phys., 181 (2020), 329-363.  doi: 10.1007/s10955-020-02583-3. [25] R. Kapral, Discrete models for chemically reacting systems, J. Math. Chem., 6 (1991), 113-163.  doi: 10.1007/BF01192578. [26] 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. [27] Y. Li, Z. Liu and W. Wang, Almost periodic solutions and stable solutions for stochastic differential equations, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 5927-5944.  doi: 10.3934/dcdsb.2019113. [28] Z. Liu and K. Sun, Almost automorphic solutions for stochastic differential equations driven by Lévy noise, J. Funct. Anal., 266 (2014), 1115-1149.  doi: 10.1016/j.jfa.2013.11.011. [29] Z. Liu and W. Wang, Favard separation method for almost periodic stochastic differential equations, J. Differential Equations, 260 (2016), 8109-8136.  doi: 10.1016/j.jde.2016.02.019. [30] Y. Lv and J. Sun, Dynamical behavior for stochastic lattice systems, Chaos Solitons Fractals, 27 (2006), 1080-1090.  doi: 10.1016/j.chaos.2005.04.089. [31] S. Ma and X. Zou, Propagation and its failure in a lattice delayed differential equation with global interaction, J. Differential Equations, 212 (2005), 129-190.  doi: 10.1016/j.jde.2004.07.014. [32] X. Ma, X.-B. Shu and J. Mao, Existence of almost periodic solutions for fractional impulsive neutral stochastic differential equations with infinite delay, Stoch. Dyn., 20 (2020), 2050003, 31 pp. doi: 10.1142/S0219493720500033. [33] X. Mao, Stochastic Differential Equations and Applications, 2$^{nd}$ edition, Horwood Publishing Limited, Chichester, 2008. doi: 10.1533/9780857099402. [34] J. Shen, J. Zhao, K. Lu and B. Wang, The Wong-Zakai approximations of invariant manifolds and foliations for stochastic evolution equations, J. Differential Equations, 266 (2019), 4568-4623.  doi: 10.1016/j.jde.2018.10.008. [35] W. Shen, Traveling waves in time periodic lattice differential equations, Nonlinear Anal., 54 (2003), 319-339.  doi: 10.1016/S0362-546X(03)00065-8. [36] B. Wang, Dynamics of systems on infinite lattices, J. Differential Equations, 221 (2006), 224-245.  doi: 10.1016/j.jde.2005.01.003. [37] X. Wang, J. Shen, K. Lu and B. Wang, Wong-Zakai approximations and random attractors for non-autonomous stochastic lattice systems, J. Differential Equations, 280 (2021), 477-516.  doi: 10.1016/j.jde.2021.01.026. [38] Y. Wang, J. Xu and P. E. Kloeden, Asymptotic behavior of stochastic lattice systems with a Caputo fractional time derivative, Nonlinear Anal., 135 (2016), 205-222.  doi: 10.1016/j.na.2016.01.020. [39] W. Yan, Y. Li and S. Ji, Random attractors for first order stochastic retarded lattice dynamical systems, J. Math. Phys., 51 (2010), 032702, 17 pp. doi: 10.1063/1.3319566. [40] C. Zhao and S. Zhou, Attractors of retarded first order lattice systems, Nonlinearity, 20 (2007), 1987-2006.  doi: 10.1088/0951-7715/20/8/010. [41] H. Zhao and Z.-H. Zheng, Random periodic solutions of random dynamical systems, J. Differential Equations, 246 (2009), 2020-2038.  doi: 10.1016/j.jde.2008.10.011. [42] 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.
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