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

TL;DR

This paper establishes integrable bounds for solutions of rough stochastic PDEs driven by Gaussian processes, enabling analysis of invariant manifolds and stability through Lyapunov spectra.

Contribution

It introduces $ ext{L}^p( ext{Ω})$-integrable bounds for solutions and linearizations of rough SPDEs, facilitating stability and invariant manifold analysis.

Findings

Existence of Lyapunov spectrum for linearized equations

Construction of local stable, unstable, and center manifolds

Conditions for local exponential stability when all Lyapunov exponents are negative

Abstract

We study semilinear rough stochastic partial differential equations as introduced in [Gerasimovi{\v{c}}s, Hairer; EJP 2019]. We provide $\mathcal{L}^p(\Omega)$-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.