Mean-square invariant manifolds for ill-posed stochastic evolution equations driven by nonlinear noise

2021/11/01 by Zonghao Li, Li, Zonghao, Caibin Zeng +3
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Dynamical Systems (math.DS) #FOS: Mathematics #Numerical methods in inverse problems #Probability (math.PR) #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2111.01043

openalex publication_date 2021/11/01 · openalex created_date 2021/11/08 · openalex updated_date 2026/07/28

Abstract

This paper discerns the invariant manifold of a class of ill-posed stochastic evolution equations driven by a nonlinear multiplicative noise. To be more precise, we establish the existence of mean-square random unstable invariant manifold and only mean-square stable invariant set. Due to the lack of the Hille-Yosida condition, we construct a modified variation of constants formula by the resolvent operator. With the price of imposing an unusual condition involving a non-decreasing map, we set up the Lyapunov-Perron method and derive the required estimates. We also emphasize that the Lyapunov-Perron map in the forward time loses the invariant due to the adaptedness, we alternatively establish the existence of mean-square random stable sets.

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