Ergodic Theorems for the Transfer Operators of Noisy Dynamical Systems

TL;DR

This paper extends classical ergodic theorems to stochastic dynamical systems with noise, proving maximal and pointwise ergodic theorems for their transfer operators on compact metric spaces.

Contribution

It introduces new ergodic theorems for transfer operators of noisy systems, generalizing deterministic ergodic results to stochastic settings.

Findings

Proved maximal ergodic theorem for noisy systems

Established pointwise ergodic theorem for transfer operators

Extended fundamental ergodic theorems to stochastic dynamical systems

Abstract

We consider stationary stochastic dynamical systems evolving on a compact metric space, by perturbing a deterministic dynamics with a random noise, added according to an arbitrary probabilistic distribution. We prove the maximal and pointwise ergodic theorems for the transfer operators associated to such systems. The results are extensions to noisy systems of some of the fundamental ergodic theorems for deterministic systems.