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Iterated invariance principle for random dynamical systems

Davor Dragicevic, Yeor Hafouta

TL;DR

The paper proves a weak iterated invariance principle for a broad class of non-uniformly expanding random dynamical systems in a quenched setting, using a martingale-coboundary decomposition to handle dependence in random environments. It first establishes CLT/LIL-type results for sums along random compositions, then builds a framework for iterated WIP by reducing to martingale limits and coboundary corrections, yielding explicit Brownian limits with possible drift terms. The work further provides a quenched homogenization result for fast-slow systems when the fast component is uniformly expanding, and demonstrates applicability to both uniformly and nonuniformly mixing random maps, including several concrete examples. The methods hinge on transfer operator decay, martingale techniques, and rough-path–type constructions, and are supported by an appendix on ergodicity and technical lemmas.

Abstract

We prove a weak iterated invariance principle for a large class of non-uniformly expanding random dynamical systems. In addition, we give a quenched homogenization result for fast-slow systems in the case when the fast component corresponds to a uniformly expanding random system. Our techniques rely on the appropriate martingale decomposition.

Iterated invariance principle for random dynamical systems

TL;DR

The paper proves a weak iterated invariance principle for a broad class of non-uniformly expanding random dynamical systems in a quenched setting, using a martingale-coboundary decomposition to handle dependence in random environments. It first establishes CLT/LIL-type results for sums along random compositions, then builds a framework for iterated WIP by reducing to martingale limits and coboundary corrections, yielding explicit Brownian limits with possible drift terms. The work further provides a quenched homogenization result for fast-slow systems when the fast component is uniformly expanding, and demonstrates applicability to both uniformly and nonuniformly mixing random maps, including several concrete examples. The methods hinge on transfer operator decay, martingale techniques, and rough-path–type constructions, and are supported by an appendix on ergodicity and technical lemmas.

Abstract

We prove a weak iterated invariance principle for a large class of non-uniformly expanding random dynamical systems. In addition, we give a quenched homogenization result for fast-slow systems in the case when the fast component corresponds to a uniformly expanding random system. Our techniques rely on the appropriate martingale decomposition.
Paper Structure (14 sections, 27 theorems, 205 equations)

This paper contains 14 sections, 27 theorems, 205 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The asymptotic variance, the CLT and the law of iterated logarithm (LIL)
  4. Martingale decomposition
  5. Iterated weak invariance principle
  6. Preliminaries
  7. Iterated weak invariance principle for martingales
  8. Iterated weak invariance principle via martingale reduction
  9. Examples
  10. Uniform decay of correlations
  11. Application to homogenization
  12. Nonuniform decay of correlations I: Non-uniformly expanding maps
  13. Nonuniform decay of correlations II: random maps with dominating expansion
  14. Appendix

Key Result

Lemma 1

For $\mathbb P$-a.e. $\omega \in \Omega$, $n\in \mathbb{N}$, $\varphi \in \mathcal{H}$ and $\psi \in L^1(\mu_{\sigma^n \omega})$, we have that

Theorems & Definitions (57)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Remark 1
  • Theorem 1
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • ...and 47 more