Non-autonomous stochastic lattice systems with Markovian switching

Dingshi Li; Yusen Lin; Zhe Pu

doi:10.3934/dcds.2022181

Article Contents

2023, Volume 43, Issue 5: 1860-1877. Doi: 10.3934/dcds.2022181

This issue Previous Article Long-time behavior of solutions to the $ M_1 $ model with boundary effect Next Article Fredholm theory of families of discrete dynamical systems and its applications to bifurcation theory

Non-autonomous stochastic lattice systems with Markovian switching

1.
School of Mathematics, Southwest Jiaotong University, Chengdu, Sichuan 610031, China

^*Corresponding author: Yusen Lin
^*Corresponding author: Yusen Lin

Received: September 2022

Revised: November 2022

Early access: December 2022

Published: May 2023

This work was supported by NSFC (11971394), Central Government Funds for Guiding Local Scientific and Technological Development (2021ZYD0010) and Fundamental Research Funds for the Central Universities (2682021ZTPY057)

Abstract / Introduction Full Text(HTML) Related Papers Cited by

Abstract

The aim of this paper is to study the dynamical behavior of non-autonomous stochastic lattice systems with Markovian switching. We first show existence of an evolution system of measures of the stochastic system. We then study the pullback (or forward) asymptotic stability in distribution of the evolution system of measures. We finally prove that any limit point of a tight sequence of an evolution system of measures of the stochastic lattice systems must be an evolution system of measures of the corresponding limiting system as the intensity of noise converges zero. In particular, when the coefficients are periodic with respect to time, we show every limit point of a sequence of periodic measures of the stochastic system must be a periodic measure of the limiting system as the noise intensity goes to zero.

Keywords:

Mathematics Subject Classification: Primary: 37L55; Secondary: 34F05, 37L30, 60H10.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

[1]	P. W. Bates, H. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems, Stochastics and Dynamics, 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: Nonlinear Phenomena, 289 (2014), 32-50. doi: 10.1016/j.physd.2014.08.004.
[3]	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.
[4]	T. Caraballo and K. Lu, Attractors for stochastic lattice dynamical systems with a multiplicative noise, Frontiers of Mathematics in China, 3 (2008), 317-335. doi: 10.1007/s11464-008-0028-7.
[5]	T. Caraballo, F. Morillas and J. Valero, Attractors of stochastic lattice dynamical systems with a multiplicative noise and non-Lipschitz nonlinearities, Journal of Differential Equations, 253 (2012), 667-693. doi: 10.1016/j.jde.2012.03.020.
[6]	Z. Chen, X. Li and B. Wang, Invariant measures of stochastic delay lattice systems, Discrete & Continuous Dynamical Systems-B, 26 (2021), 3235-3269. doi: 10.3934/dcdsb.2020226.
[7]	G. Da Prato and A. Debussche, 2D stochastic navier-stokes equations with a time-periodic forcing term, Journal of Dynamics and Differential Equations, 20 (2008), 301-335. doi: 10.1007/s10884-007-9074-1.
[8]	G. Da Prato and M. Röckner, Dissipative stochastic equations in Hilbert space with time dependent coefficients, Rendiconti Lincei-Matematica e Applicazioni, 17 (2006), 397-403. doi: 10.4171/RLM/476.
[9]	N. H. Du, N. H. Dang and N. T. Dieu, On stability in distribution of stochastic differential delay equations with markovian switching, Systems & Control Letters, 65 (2014), 43-49. doi: 10.1016/j.sysconle.2013.12.006.
[10]	X. Han, P. E. Kloeden and B. Usman, Long term behavior of a random hopfield neural lattice model, Communications on Pure & Applied Analysis, 18 (2019), 809-824. doi: 10.3934/cpaa.2019039.
[11]	D. Li, B. Wang and X. Wang, Limiting behavior of invariant measures of stochastic delay lattice systems, Journal of Dynamics and Differential Equations, 34 (2022), 1453-1487. doi: 10.1007/s10884-021-10011-7.
[12]	D. Li, B. Wang and X. Wang, Periodic measures of stochastic delay lattice systems, Journal of Differential Equations, 272 (2021), 74-104. doi: 10.1016/j.jde.2020.09.034.
[13]	R. Liu and K. Lu, Statistical properties of 2D stochastic Navier-stokes equations with time-periodic forcing and degenerate stochastic forcing, preprint, 2021, arXiv: 2105.00598.
[14]	X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, 2006. doi: 10.1142/p473.
[15]	M. Sui, Y. Wang, X. Han and P. E. Kloeden, Random recurrent neural networks with delays, Journal of Differential Equations, 269 (2020), 8597-8639. doi: 10.1016/j.jde.2020.06.008.
[16]	B. Wang, Dynamics of stochastic reaction–diffusion lattice systems driven by nonlinear noise, Journal of Mathematical Analysis and Applications, 477 (2019), 104-132. doi: 10.1016/j.jmaa.2019.04.015.
[17]	B. Wang and R. Wang, Asymptotic behavior of stochastic schrödinger lattice systems driven by nonlinear noise, Stochastic Analysis and Applications, 38 (2020), 213-237. doi: 10.1080/07362994.2019.1679646.
[18]	R. Wang, T. Caraballo and N. Huy Tuan, Asymptotic stability of evolution systems of probability measures for nonautonomous stochastic systems: Theoretical results and applications, preprint, 2022, arXiv: 2203.13039.
[19]	R. Wang and B. Wang, Random dynamics of $p$-laplacian lattice systems driven by infinite-dimensional nonlinear noise, Stochastic Processes and their Applications, 130 (2020), 7431-7462. doi: 10.1016/j.spa.2020.08.002.
[20]	X. Wang, K. Lu and B. Wang, Exponential stability of non-autonomous stochastic delay lattice systems with multiplicative noise, Journal of Dynamics and Differential Equations, 28 (2016), 1309-1335. doi: 10.1007/s10884-015-9448-8.
[21]	C. Yuan and X. Mao, Asymptotic stability in distribution of stochastic differential equations with markovian switching, Stochastic Processes and their Applications, 103 (2003), 277-291. doi: 10.1016/S0304-4149(02)00230-2.
[22]	C. Yuan, J. Zou and X. Mao, Stability in distribution of stochastic differential delay equations with markovian switching, Systems & Control Letters, 50 (2003), 195-207. doi: 10.1016/S0167-6911(03)00154-3.