Physical measures for nonlinear random walks on interval

TL;DR

This paper studies nonlinear random walks on the interval, characterizing their ergodic measures, and shows they have negative Lyapunov exponents, with a detailed analysis of attractors and physical measures in related skew product systems.

Contribution

It provides a geometric characterization of ergodic stationary measures for nonlinear random walks and analyzes the structure of attractors and physical measures in step skew product systems.

Findings

All ergodic stationary measures have negative Lyapunov exponents.

Existence of finitely many alternating attractors and repellers with a sharp upper bound.

Attractors support ergodic hyperbolic physical measures.

Abstract

A one-dimensional confined Nonlinear Random Walk is a tuple of $N$ diffeomorphisms of the unit interval driven by a probabilistic Markov chain. For generic such walks, we obtain a geometric characterization of their ergodic stationary measures and prove that all of them have negative Lyapunov exponents. These measures appear to be probabilistic manifestations of physical measures for certain deterministic dynamical systems. These systems are step skew products over transitive subshifts of finite type (topological Markov chains) with the unit interval fiber. For such skew products, we show there exist only finite collection of alternating attractors and repellers; we also give a sharp upper bound for their number. Each of them is a graph of a continuous map from the base to the fiber defined almost everywhere w.r.t. any ergodic Markov measure in the base. The orbits starting between the adjacent attractor and repeller tend to the attractor as $t \to +\infty$, and to the repeller as $t \to -\infty$. The attractors support ergodic hyperbolic physical measures.