# American Institute of Mathematical Sciences

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  October 2019 Early access  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
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