Studies on invariant measures of fractional stochastic delay Ginzburg-Landau equations on $ \mathbb{R}^n $

Hong Lu; Linlin Wang; Mingji Zhang; Hong Lu; Linlin Wang; Mingji Zhang

doi:10.3934/mbe.2024241

Mathematical Biosciences and Engineering

2024, Volume 21, Issue 4: 5456-5498. doi: 10.3934/mbe.2024241

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Studies on invariant measures of fractional stochastic delay Ginzburg-Landau equations on $ \mathbb{R}^n $

1.
School of Mathematics and Statistics, Shandong University, Weihai 264209, China
2.
Department of Mathematics, New Mexico Institute of Mining and Technology, Socorro NM 87801, USA

Received: 15 December 2023 Revised: 26 February 2024 Accepted: 29 February 2024 Published: 15 March 2024

This paper is concerned with invariant measures of fractional stochastic delay Ginzburg-Landau equations on the entire space $ \mathbb{R}^n $. We first derive the uniform estimates and the mean-square uniform smallness of the tails of solutions in corresponding space. Then we deduce the weak compactness of a set of probability distributions of the solutions applying the Ascoli-Arzel$ \grave{a} $. We finally prove the existence of invariant measures by applying Krylov-Bogolyubov's method.
- stochastic delay equation,
- unbounded domain,
- fractional Ginzburg-Landau equation,
- invariant measure
Citation: Hong Lu, Linlin Wang, Mingji Zhang. Studies on invariant measures of fractional stochastic delay Ginzburg-Landau equations on $ \mathbb{R}^n $[J]. Mathematical Biosciences and Engineering, 2024, 21(4): 5456-5498. doi: 10.3934/mbe.2024241

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Abstract

This paper is concerned with invariant measures of fractional stochastic delay Ginzburg-Landau equations on the entire space $ \mathbb{R}^n $. We first derive the uniform estimates and the mean-square uniform smallness of the tails of solutions in corresponding space. Then we deduce the weak compactness of a set of probability distributions of the solutions applying the Ascoli-Arzel$ \grave{a} $. We finally prove the existence of invariant measures by applying Krylov-Bogolyubov's method.

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