Finitude of physical measures for random maps

TL;DR

This paper investigates the existence and finiteness of physical measures for random dynamical systems, providing characterizations, examples, and practical conditions for their finitude on Polish spaces and manifolds.

Contribution

It introduces a hierarchy of random maps based on their physical measures and offers criteria for finiteness, including for continuous maps on manifolds with absolutely continuous transitions.

Findings

Finitely many physical measures exist for continuous random maps on compact Riemannian manifolds.

Hierarchy of random maps characterized by their associated Markov operators.

Practical conditions established for the finiteness of physical measures.

Abstract

For random compositions of independent and identically distributed measurable maps on a Polish space, we study the existence and finitude of absolutely continuous ergodic stationary probability measures (which are, in particular, physical measures) whose basins of attraction cover the whole space almost everywhere. We characterize and hierarchize such random maps in terms of their associated Markov operators, as well as show the difference between classes in the hierarchy by plenty of examples, including additive noise, multiplicative noise, and iterated function systems. We also provide sufficient practical conditions for a random map to belong to these classes. For instance, we establish that any continuous random map on a compact Riemannian manifold with absolutely continuous transition probability has finitely many physical measures whose basins of attraction cover Lebesgue almost all the manifold.