# Equivalence of physical and SRB measures in random dynamical systems

## TL;DR

This paper provides a geometric proof demonstrating that, under mild conditions, sample measures of random diffeomorphisms are SRB measures, establishing the equivalence of physical and SRB measures in random dynamical systems.

## Contribution

It offers a new geometric perspective on the equivalence of physical and SRB measures specifically for random dynamical systems, extending prior results.

## Key findings

- Sample measures of random diffeomorphisms are SRB measures.
- Sample measures are limits of forward images of stationary measures.
- Physical and SRB measures are equivalent in the random setting.

## Abstract

We give a geometric proof, offering a new and quite different perspective on an earlier result of Ledrappier and Young on random transformations. We show that under mild conditions, sample measures of random diffeomorphisms are SRB measures. As sample measures are the limits of forward images of stationary measures, they can be thought of as the analog of physical measures for deterministic systems. Our results thus show the equivalence of physical and SRB measures in the random setting, a hoped-for scenario that is not always true for deterministic maps.