Periodic measures of stochastic delay lattice systems

Abstract The periodic measures of the stochastic delay reaction-diffusion lattice systems are investigated. Under a general condition, we prove the existence of periodic measures when the time-dependent terms of the system are periodic in time. Under further assumptions on the nonlinear terms, we show the set of all periodic measures of the perturbed system is weakly compact. Finally, we prove every limit point of a sequence of periodic measures of the stochastic delay system must be a periodic measure of the limiting system as the noise intensity or the time delay goes to zero.

[1]  Zhang Chen,et al.  Invariant measures of stochastic delay lattice systems , 2021, Discrete & Continuous Dynamical Systems - B.

[2]  Attractors for discrete nonlinear Schrödinger equation with delay , 2010 .

[3]  Peter E. Kloeden,et al.  Asymptotic Behavior of a Nonautonomous p-Laplacian Lattice System , 2016, Int. J. Bifurc. Chaos.

[4]  Pullback attractors for a class of nonlinear lattices with delays , 2015 .

[5]  Xuerong Mao,et al.  Stability in distribution of stochastic differential delay equations with Markovian switching , 2003, Syst. Control. Lett..

[6]  Shengfan Zhou,et al.  Existence and upper semicontinuity of attractors fornon-autonomous stochastic lattice systems with random coupledcoefficients , 2016 .

[7]  J. U. Kim.  PERIODIC AND INVARIANT MEASURES FOR STOCHASTIC WAVE EQUATIONS , 2004 .

[8]  Peter W. Bates,et al.  A Discrete Convolution Model¶for Phase Transitions , 1999 .

[9]  T. Caraballo,et al.  On differential equations with delay in Banach spaces and attractors for retarded lattice dynamical systems , 2013 .

[10]  Weiping Yan,et al.  Random attractors for first order stochastic retarded lattice dynamical systems , 2010 .

[11]  Tomás Caraballo,et al.  Attractors of stochastic lattice dynamical systems with a multiplicative noise and non-Lipschitz nonlinearities , 2012 .

[12]  Shengfan Zhou,et al.  Attractors of retarded first order lattice systems , 2007 .

[13]  Xinfu Chen,et al.  Traveling Waves of Bistable Dynamics on a Lattice , 2003, SIAM J. Math. Anal..

[14]  Xiaohu Wang,et al.  Exponential Stability of Non-Autonomous Stochastic Delay Lattice Systems with Multiplicative Noise , 2016 .

[15]  Xiaoying Han,et al.  Non-autonomous lattice systems with switching effects and delayed recovery , 2016 .

[16]  S. Mohammed.  Stochastic functional differential equations , 1984 .

[17]  Xiaoying Han,et al.  Random attractors for stochastic lattice dynamical systems in weighted spaces , 2011 .

[18]  Bixiang Wang,et al.  Dynamics of stochastic reaction–diffusion lattice systems driven by nonlinear noise , 2019, Journal of Mathematical Analysis and Applications.

[19]  Bixiang Wang,et al.  Dynamics of systems on infinite lattices , 2006 .

[20]  PERIODIC SOLUTIONS OF STOCHASTIC DELAY DIFFERENTIAL EQUATIONS AND APPLICATIONS TO LOGISTIC EQUATION AND NEURAL NETWORKS , 2013 .

[21]  Z. Brzeźniak,et al.  Invariant measures for stochastic nonlinear beam and wave equations , 2016 .

[22]  R. Khasminskii.  Stochastic Stability of Differential Equations , 1980 .

[23]  Peter W. Bates,et al.  Attractors for Lattice Dynamical Systems , 2001, Int. J. Bifurc. Chaos.

[24]  Wolf-Jürgen Beyn,et al.  Attractors of Reaction Diffusion Systems on Infinite Lattices , 2003 .

[25]  B. G. Zhang,et al.  On the periodic solution of N–Dimensional Stochastic Population Models , 2000 .

[26]  George Yin,et al.  Ergodicity for functional stochastic differential equations and applications , 2014 .

[27]  Bixiang Wang.  Dynamics of fractional stochastic reaction-diffusion equations on unbounded domains driven by nonlinear noise , 2019 .

[28]  Daoyi Xu,et al.  Existence theorems for periodic Markov process and stochasticfunctional differential equations , 2009 .

[29]  Caidi Zhao,et al.  Global attractors for the discrete Klein–Gordon–Schrödinger type equations , 2014 .

[30]  Yuecai Han,et al.  Periodic solutions of Fokker–Planck equations , 2017 .

[31]  Liguang Xu,et al.  Existence and uniqueness theorems for periodic Markov process and applications to stochastic functional differential equations , 2018, Journal of Mathematical Analysis and Applications.

[32]  Shui-Nee Chow,et al.  Traveling Waves in Lattice Dynamical Systems , 1998 .

[33]  G. Yin,et al.  Stochastic functional differential equations with infinite delay: Existence and uniqueness of solutions, solution maps, Markov properties, and ergodicity , 2017 .

[34]  X. Mao,et al.  Stochastic Differential Equations and Applications , 1998 .

[35]  Michael Scheutzow,et al.  Invariant measures for stochastic functional differential equations , 2017, 1703.05120.

[36]  Daoyi Xu,et al.  Random attractors for second-order stochastic lattice dynamical systems☆ , 2010 .

[37]  Chengjian Zhang,et al.  The attractors for 2nd-order stochastic delay lattice systems , 2016 .

[38]  G. Prato,et al.  Periodic and almost periodic solutions for semilinear stochastic equations , 1995 .

[39]  J. U. Kim.  On the stochastic Benjamin–Ono equation , 2006 .

[40]  Xuerong Mao,et al.  Stability in distribution of stochastic functional differential equations , 2019, Syst. Control. Lett..

[41]  P. Bates,et al.  Attractors of non-autonomous stochastic lattice systems in weighted spaces , 2014 .

[42]  M. Ondreját,et al.  Invariant measure for the stochastic Navier-Stokes equations in unbounded 2D domains , 2015, 1502.02637.

[43]  P. Bates,et al.  ATTRACTORS FOR STOCHASTIC LATTICE DYNAMICAL SYSTEMS , 2006 .