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Lipschitz regularity of the invariant measure of random dynamical systems

Davi Lima, Rafael Lucena

TL;DR

The work addresses regularity and mixing properties of invariant measures for a class of random dynamical systems modeled as skew-products with contracting fibers. By constructing anisotropic spaces built from a Wasserstein-like metric and analyzing the transfer operator near its fixed point, the authors prove a spectral gap and a unique invariant measure in the strong space. This yields exponential convergence to equilibrium and exponential decay of correlations for Lipschitz observables, with Lipschitz regularity of the invariant measure’s disintegration along stable fibers. The results provide a robust statistical framework for RDS and related IFS settings, highlighting precise quantitative rates and structural regularity that can support further probabilistic limit theorems and stability analyses.

Abstract

In this article we derive a regularity result for the disintegration of the invariant measure associated to a class of Random Dynamical Systems - RDS. The results of this work are obtained by constructing a suitable anisotropic normed space defined by the Wasserstein-Kantorovich-like metric and understanding the dynamics of the associated transfer operator in a neighborhood of its fixed point. Precisely, we employ functional analytic techniques to demonstrate a spectral gap for its action on suitable spaces of signed measures. We apply this analysis to prove an exponential decay of correlation statement for Lipschitz observables and statistical properties of the RDS.

Lipschitz regularity of the invariant measure of random dynamical systems

TL;DR

The work addresses regularity and mixing properties of invariant measures for a class of random dynamical systems modeled as skew-products with contracting fibers. By constructing anisotropic spaces built from a Wasserstein-like metric and analyzing the transfer operator near its fixed point, the authors prove a spectral gap and a unique invariant measure in the strong space. This yields exponential convergence to equilibrium and exponential decay of correlations for Lipschitz observables, with Lipschitz regularity of the invariant measure’s disintegration along stable fibers. The results provide a robust statistical framework for RDS and related IFS settings, highlighting precise quantitative rates and structural regularity that can support further probabilistic limit theorems and stability analyses.

Abstract

In this article we derive a regularity result for the disintegration of the invariant measure associated to a class of Random Dynamical Systems - RDS. The results of this work are obtained by constructing a suitable anisotropic normed space defined by the Wasserstein-Kantorovich-like metric and understanding the dynamics of the associated transfer operator in a neighborhood of its fixed point. Precisely, we employ functional analytic techniques to demonstrate a spectral gap for its action on suitable spaces of signed measures. We apply this analysis to prove an exponential decay of correlation statement for Lipschitz observables and statistical properties of the RDS.

Paper Structure

This paper contains 23 sections, 32 theorems, 158 equations.

Table of Contents

  1. Introduction
  2. Settings
  3. Sub-shifts of finite type
  4. Contracting Fiber Maps
  5. Properties of $G$
  6. Statements of the Main Results
  7. Preliminares
  8. The $\mathcal{L}^{\infty}$ and $S^{\infty}$ spaces
  9. Transfer operator associated to $F$
  10. Basic properties of the norms and convergence to equilibrium
  11. Convergence to equilibrium
  12. ${L}^{\infty }$ norms
  13. Spectral gap
  14. Consequences
  15. Lipschitz regularity of the disintegration of the invariant measure
  16. ...and 8 more sections

Key Result

Theorem A

Suppose that $F$ satisfies (G1). Then, there exists a unique $F$-invariant probability measure ${ \if@compatibility \mathchar"0116 {} \mathchar"0116 } _0 \in S^{\infty}$ which satisfies $||{ \if@compatibility \mathchar"0116 {} \mathchar"0116 }_0||_\infty =1$ and $||{ \if@compatibility

Theorems & Definitions (79)

  • Example 1
  • Theorem A
  • Theorem B: Spectral gap
  • Theorem C
  • Theorem D
  • Theorem 3.1
  • Lemma 3.2
  • Definition 1
  • Lemma 3.3
  • proof
  • ...and 69 more