Wong-Zakai approximations of stochastic lattice systems driven by long-range interactions and multiplicative white noises

Yiju Chen; Xiaohu Wang; Kenan Wu

doi:10.3934/dcdsb.2022113

Article Contents

2023, Volume 28, Issue 2: 1092-1115. Doi: 10.3934/dcdsb.2022113

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Wong-Zakai approximations of stochastic lattice systems driven by long-range interactions and multiplicative white noises

Department of Mathematics, Sichuan University, Chengdu, Sichuan 610065, China

^*Corresponding author: Kenan Wu
^*Corresponding author: Kenan Wu

Received: November 30, 2021

Revised: February 28, 2022

Early access: June 2022

Published: February 2023

This work was supported by NSFC (11871049 and 12090013) and Young crop project of Sichuan University (2020SCUNL111)

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Abstract

In this paper, we study the Wong-Zakai approximations of a stochastic lattice differential equation with long-range interactions and multiplicative white noise at each node. We first prove the existence and uniqueness of pullback random attractors for lattice system driven by multiplicative white noises as well as the corresponding Wong-Zakai approximate system. Then, we prove the convergence of solutions and the upper semicontinuity of random attractors for the Wong-Zakai approximate system as the size of approximation approaches zero.

Keywords:

Stochastic lattice differential equation,
long-range interaction,
Wong-Zakai approximation,
upper semicontinuity.

Mathematics Subject Classification: Primary: 37L55, 37L60, 35B41; Secondary: 60H15, 35B40.

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