Comparison theorems for invariant measures of random dynamical systems

TL;DR

This paper introduces comparison theorems that simplify estimating invariant measures in complex random dynamical systems composed of multiple deterministic maps and probability densities.

Contribution

It provides fundamental comparison theorems that facilitate the estimation of invariant measures in complicated random maps, including those with indifferent fixed points or unbounded derivatives.

Findings

Comparison theorems ease invariant measure estimation

Application to systems with indifferent fixed points

Application to systems with unbounded derivatives

Abstract

We study a random dynamical system such that one transformation is randomly selected from a family of transformations and then applied on each iteration. For such random dynamical systems, we consider estimates of absolutely continuous invariant measures. Since the random dynamical systems are made by complicated compositions of many deterministic maps and probability density functions, it is difficult to estimate the invariant measures. To get rid of this difficulty, we present fundamental comparison theorems which make easier the estimates of invariant measures of random maps. We also demonstrate how to apply the comparison theorems to random maps with indifferent fixed points and/or with unbounded derivatives.