An integrable bound for rough stochastic partial differential equations with applications to invariant manifolds and stability

2023/07/04 by Mazyar Ghani Varzaneh, Sebastian Riedel, Varzaneh, Mazyar Ghani +1
Engineering · #FOS: Mathematics #Probability (math.PR) #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2307.01679

openalex publication_date 2023/07/04 · openalex created_date 2023/07/07 · openalex updated_date 2026/07/28

Abstract

We study semilinear rough stochastic partial differential equations as introduced in [Gerasimovičs, Hairer; EJP 2019]. We provide L^p(Ω)-integrable a priori bounds for the solution and its linearization in case the equation is driven by a suitable Gaussian process. Using the Multiplicative Ergodic Theorem for Banach spaces, we can deduce the existence of a Lyapunov spectrum for the linearized equation around stationary points. The existence of local stable, unstable, and center manifolds around stationary points is also provided. In the case where all Lyapunov exponents are negative, local exponential stability can be deduced. We illustrate our findings with several examples.

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