Random invariant densities for Markov operator cocycles and random mean ergodic theorem

TL;DR

This paper investigates the existence of random invariant densities and establishes a mean ergodic theorem for Markov operator cocycles in random dynamical systems, linking weak precompactness to strong convergence.

Contribution

It provides necessary and sufficient conditions for random invariant densities and proves a mean ergodic theorem applicable to generalized linear operator cocycles.

Findings

Conditions for existence of random invariant densities

Weak precompactness implies strong convergence

Applicability to quenched random dynamical systems

Abstract

In the present paper, we consider random invariant densities and the mean ergodic theorem for Markov operator cocycles which are applicable to quenched type random dynamical systems. We give necessary and sufficient conditions for the existence of random invariant densities for Markov operator cocycles and establish the mean ergodic theorem for generalized linear operator cocycles over a weakly sequentially complete Banach space. The advantage of the result is that we show the implication of weak precompactness for almost every environment to strong convergence in the global sense.