Stability and limit theorems in random dynamical systems

TL;DR

This paper studies the statistical stability and limit theorems of random dynamical systems driven by subshifts, introducing a flexible operator approach that yields explicit stability bounds, decay of correlations, and a CLT.

Contribution

It develops a novel operator method based on Lipschitz regularity and Wasserstein metrics to analyze stability and limit laws in random dynamical systems, bypassing classical anisotropic Banach space techniques.

Findings

Proves explicit quantitative stability of invariant measures under perturbations.

Establishes exponential decay of correlations for Lipschitz observables.

Proves the Central Limit Theorem for Birkhoff averages in this setting.

Abstract

The robust statistical description of dynamical systems under perturbations is a central problem in ergodic theory. In this paper, we investigate the statistical properties of skew-product maps driven by a subshift of finite type with contracting fiber maps, a setting that naturally encompasses Iterated Function Systems (IFS) and Random Dynamical Systems (RDS). Diverging from the classical perturbative frameworks that rely on the compact embedding of anisotropic Banach spaces, we employ a flexible operator approach based on the Lipschitz regularity of the invariant measure's disintegrations with respect to the Wasserstein metric. Our main results are threefold: first, we prove the quantitative statistical stability of the unique invariant measure under admissible deterministic perturbations, obtaining an explicit modulus of continuity of the form $O(R(\delta) \log \delta)$. Second, we establish the exponential decay of correlations on new pair of spaces of observables. Finally, leveraging this exponential decay and Gordin's method, we prove the Central Limit Theorem for the fluctuations of Birkhoff averages of Lipschitz observables.