Multiplicative ergodic theorem for a non-irreducible random dynamical system

TL;DR

This paper proves a multiplicative ergodic theorem for certain non-irreducible infinite-dimensional random dynamical systems, with applications to PDEs like Navier-Stokes, providing new statistical insights including large deviations and pressure function analyticity.

Contribution

It establishes a multiplicative ergodic theorem with exponential convergence for non-irreducible systems, extending ergodic theory to broader classes of PDEs.

Findings

Proves a multiplicative ergodic theorem with exponential rate.

Derives large deviations principle for occupation measures.

Shows analyticity of the pressure function in non-irreducible settings.

Abstract

We study the asymptotic properties of the trajectories of a discrete-time random dynamical system in an infinite-dimensional Hilbert space. Under some natural assumptions on the model, we establish a multiplica-tive ergodic theorem with an exponential rate of convergence. The assumptions are satisfied for a large class of parabolic PDEs, including the 2D Navier--Stokes and complex Ginzburg--Landau equations perturbed by a non-degenerate bounded random kick force. As a consequence of this er-godic theorem, we derive some new results on the statistical properties of the trajectories of the underlying random dynamical system. In particular , we obtain large deviations principle for the occupation measures and the analyticity of the pressure function in a setting where the system is not irreducible. The proof relies on a refined version of the uniform Feller property combined with some contraction and bootstrap arguments.