Averaging principle for stochastic real Ginzburg-Landau equation driven by $ \alpha $-stable process

Xiaobin Sun; Jianliang Zhai

doi:10.3934/cpaa.2020063

Article Contents

2020, Volume 19, Issue 3: 1291-1319. Doi: 10.3934/cpaa.2020063

This issue Previous Article Convergence of lacunary SU(1, 1)-valued trigonometric products Next Article Admissibility and polynomial dichotomies for evolution families

Averaging principle for stochastic real Ginzburg-Landau equation driven by $ \alpha $-stable process

Xiaobin Sun^1, , and
Jianliang Zhai^2,

1.
School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, 221116, China
2.
Key Laboratory of Wu Wen-Tsun Mathematics, CAS, School of Mathematical Science, University of Science and Technology of China, Hefei, 230026, China

^* Corresponding author
^* Corresponding author

Received: February 28, 2019

Revised: July 31, 2019

Published: February 2020

Xiaobin Sun is supported by the National Natural Science Foundation of China (11601196, 11771187, 11931004), the NSF of Jiangsu Province (No. BK20160004) and the Priority Academic Program Development of Jiangsu Higher Education Institutions. Jianliang Zhai is supported by the National Natural Science Foundation of China (11431014, 11671372, 11721101), the Fundamental Research Funds for the Central Universities (No. WK0010450002, WK3470000008), Key Research Program of Frontier Sciences, CAS, No: QYZDB-SSW-SYS009, School Start-up Fund (USTC) KY0010000036

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Abstract

In this paper, we study a system of stochastic partial differential equations with slow and fast time-scales, where the slow component is a stochastic real Ginzburg-Landau equation and the fast component is a stochastic reaction-diffusion equation, the system is driven by cylindrical $ \alpha $-stable process with $ \alpha\in (1, 2) $. Using the classical Khasminskii approach based on time discretization and the techniques of stopping times, we show that the slow component strong converges to the solution of the corresponding averaged equation under some suitable conditions.

Keywords:

Stochastic real Ginzburg-Landau equation,
averaging principle,
ergodicity,
invariant measure,
strong convergence,
cylindrical $ \alpha $-stable.

Mathematics Subject Classification: Primary: 35R60; Secondary: 60H15.

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