Mean-square random invariant manifolds for stochastic differential equations

Bixiang Wang

doi:10.3934/dcds.2020324

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2021, Volume 41, Issue 3: 1449-1468. Doi: 10.3934/dcds.2020324

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Mean-square random invariant manifolds for stochastic differential equations

Bixiang Wang

Department of Mathematics, New Mexico Institute of Mining and Technology, Socorro, NM 87801, USA

Received: January 31, 2020

Early access: September 2020

Published: March 2021

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Abstract

We develop a theory of mean-square random invariant manifolds for mean-square random dynamical systems generated by stochastic differential equations. This theory is applicable to stochastic partial differential equations driven by nonlinear noise. The existence of mean-square random invariant unstable manifolds is proved by the Lyapunov-Perron method based on a backward stochastic differential equation involving the conditional expectation with respect to a filtration. The existence of mean-square random stable invariant sets is also established but the existence of mean-square random stable invariant manifolds remains open.

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Mathematics Subject Classification: Primary: 37D10, 60H15; Secondary: 35B42, 35L25, 37L55.

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