# Hilbert Space Lyapunov Exponent stability

## TL;DR

This paper investigates how small Gaussian noise affects the stability of Lyapunov exponents and Oseledets spaces in cocycles of compact operators on infinite-dimensional Hilbert spaces, establishing convergence results.

## Contribution

It provides the first known results on the stability of Lyapunov exponents and Oseledets spaces under noise for infinite-dimensional operator cocycles.

## Key findings

- Lyapunov exponents converge as noise diminishes
- Oseledets spaces converge in probability
- Addresses challenges unique to infinite-dimensional spaces

## Abstract

We study cocycles of compact operators acting on a separable Hilbert space, and investigate the stability of the Lyapunov exponents and Oseledets spaces when the operators are subjected to additive Gaussian noise. We show that as the noise is shrunk to 0, the Lyapunov exponents of the perturbed cocycle converge to those of the unperturbed cocycle; and the Oseledets spaces converge in probability to those of the unperturbed cocycle. This is, to our knowledge, the first result of this type with cocycles taking values in operators on infinite-dimensional spaces. The infinite dimensionality gives rise to a number of substantial difficulties that are not present in the finite-dimensional case.