# American Institute of Mathematical Sciences

March  2021, 41(3): 1449-1468. doi: 10.3934/dcds.2020324

## Mean-square random invariant manifolds for stochastic differential equations

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

Received  February 2020 Published  September 2020

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.

Citation: Bixiang Wang. Mean-square random invariant manifolds for stochastic differential equations. Discrete & Continuous Dynamical Systems, 2021, 41 (3) : 1449-1468. doi: 10.3934/dcds.2020324
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