Invariant measures of stochastic delay lattice systems

Zhang Chen; Xiliang Li; Bixiang Wang

doi:10.3934/dcdsb.2020226

Article Contents

2021, Volume 26, Issue 6: 3235-3269. Doi: 10.3934/dcdsb.2020226

This issue Previous Article Reversible polynomial Hamiltonian systems of degree 3 with nilpotent saddles Next Article Global well-posedness of a 3D Stokes-Magneto equations with fractional magnetic diffusion

Invariant measures of stochastic delay lattice systems

1.
School of Mathematics, Shandong University, Jinan 250100, China
2.
School of Mathematics and Information Science, Shandong Technology and Business University, Yantai, Shandong 264005, China
3.
Department of Mathematics, New Mexico Institute of Mining and Technology, Socorro, NM 87801, USA

^* Corresponding author: Bixiang Wang
^* Corresponding author: Bixiang Wang

Received: December 31, 2019

Revised: April 30, 2020

Early access: July 2020

Published: June 2021

Zhang Chen is partially supported by NNSF of China grant 11471190, 11971260, NSF of Shandong Province grant ZR2014AM002. Xiliang Li is partially supported by NNSF of China grant 11971273 and NSF of Shandong Province grant ZR2018MA004

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This paper is concerned with the existence and uniqueness of invariant measures for infinite-dimensional stochastic delay lattice systems defined on the entire integer set. For Lipschitz drift and diffusion terms, we prove the existence of invariant measures of the systems by showing the tightness of a family of probability distributions of solutions in the space of continuous functions from a finite interval to an infinite-dimensional space, based on the idea of uniform tail-estimates, the technique of diadic division and the Arzela-Ascoli theorem. We also show the uniqueness of invariant measures when the Lipschitz coefficients of the nonlinear drift and diffusion terms are sufficiently small.

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

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