Ergodic invariant measures of order preserving random dynamical systems on the real line are Dirac measures

TL;DR

This paper proves that ergodic invariant measures for order preserving one-sided time random dynamical systems on the real line are also Dirac measures, extending known results from two-sided systems.

Contribution

It extends the known characterization of ergodic invariant measures from two-sided to one-sided time order preserving RDS on the real line.

Findings

Ergodic invariant measures are Dirac for one-sided time RDS.

Extension of previous results from two-sided to one-sided systems.

Reinforces the uniqueness of ergodic measures in this setting.

Abstract

It is well known that ergodic invariant measures for order preserving two-sided time random dynamical systems(RDS) on the real line $\mathbb R$ are Dirac. In the present note this is shown to hold also for one-sided time RDS.