Convergence and center manifolds for differential equations driven by colored noise

Jun Shen; Kening Lu; Bixiang Wang

doi:10.3934/dcds.2019196

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2019, Volume 39, Issue 8: 4797-4840. Doi: 10.3934/dcds.2019196

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Convergence and center manifolds for differential equations driven by colored noise

1.
School of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China
2.
Department of Mathematics, Brigham Young University, Provo, Utah 84602, USA
3.
Department of Mathematics New Mexico Institute of Mining and Technology, Socorro, NM 87801, USA

^* Corresponding author: Jun Shen, junshen85@163.com
^* Corresponding author: Jun Shen, junshen85@163.com

Received: November 2018

Published: May 2019

This work was supported by NSFC (11501549, 11331007, 11831012), NSF (1413603), the Fundamental Research Funds for the Central Universities (YJ201646) and International Visiting Program for Excellent Young Scholars of SCU.

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Abstract

In this paper, we study the convergence and pathwise dynamics of random differential equations driven by colored noise. We first show that the solutions of the random differential equations driven by colored noise with a nonlinear diffusion term uniformly converge in mean square to the solutions of the corresponding Stratonovich stochastic differential equation as the correlation time of colored noise approaches zero. Then, we construct random center manifolds for such random differential equations and prove that these manifolds converge to the random center manifolds of the corresponding Stratonovich equation when the noise is linear and multiplicative as the correlation time approaches zero.

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Mathematics Subject Classification: Primary: 60H10; Secondary: 37D10, 37H10.

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