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Regularity of the variance in quenched CLT for random intermittent dynamical systems

Davor Dragičević, Juho Leppänen

TL;DR

This work analyzes random dynamical systems generated by LSV maps without base mixing and proves a quenched central limit theorem for ergodic random compositions, with an explicit variance formula. It develops a rigorous perturbation theory for the limit variance, establishing both continuity and differentiability of the variance with respect to perturbations of the random dynamics, and provides an explicit derivative formula under a spectral-gap-type regime. The methodology combines statistical stability and linear response for random intermittent maps with cone-analytic techniques, yielding precise control of density perturbations and transfer-operator differences. The results extend quenched limit theorems to non-mixing bases, offer sharp linear-response formulas, and identify parameter regimes and observable classes where variance regularity holds, contributing to the understanding of variance regularity in quenched limit theorems for nonuniformly expanding random systems.

Abstract

We study random dynamical systems composed of LSV maps with varying parameters, without any mixing assumptions on the base space of random dynamics. We establish a quenched central limit theorem and identify conditions under which the associated limit variance varies continuously and differentiably with respect to perturbations of the random dynamics. Our arguments rely on recent results on statistical stability and linear response for random intermittent maps established in Dragicevic et al. (J. Lond. Math. Soc. 111 (2025), e70150).

Regularity of the variance in quenched CLT for random intermittent dynamical systems

TL;DR

This work analyzes random dynamical systems generated by LSV maps without base mixing and proves a quenched central limit theorem for ergodic random compositions, with an explicit variance formula. It develops a rigorous perturbation theory for the limit variance, establishing both continuity and differentiability of the variance with respect to perturbations of the random dynamics, and provides an explicit derivative formula under a spectral-gap-type regime. The methodology combines statistical stability and linear response for random intermittent maps with cone-analytic techniques, yielding precise control of density perturbations and transfer-operator differences. The results extend quenched limit theorems to non-mixing bases, offer sharp linear-response formulas, and identify parameter regimes and observable classes where variance regularity holds, contributing to the understanding of variance regularity in quenched limit theorems for nonuniformly expanding random systems.

Abstract

We study random dynamical systems composed of LSV maps with varying parameters, without any mixing assumptions on the base space of random dynamics. We establish a quenched central limit theorem and identify conditions under which the associated limit variance varies continuously and differentiably with respect to perturbations of the random dynamics. Our arguments rely on recent results on statistical stability and linear response for random intermittent maps established in Dragicevic et al. (J. Lond. Math. Soc. 111 (2025), e70150).
Paper Structure (11 sections, 20 theorems, 182 equations)

This paper contains 11 sections, 20 theorems, 182 equations.

Table of Contents

  1. Introduction
  2. Convention.
  3. Quenched central limit theorem for random LSV maps
  4. Continuity of the variance
  5. Differentiability of the variance
  6. Differentiability of the first term in \ref{['Sigmavar']}
  7. The differentiability of the second term in \ref{['Sigmavar']}
  8. A series of auxiliary lemmas
  9. An improvement of Theorem \ref{['diffvar']} for a special class of observables
  10. Appendix A
  11. Appendix B

Key Result

Proposition 1

There exists $C_\alpha>0$ such that for $\mathbb P$-a.e. $\omega \in \Omega$, $n\in \mathbb N$, and $\varphi \in C^1[0, 1]$, $h\in \mathcal{C}_*$ with $\int_0^1 \varphi h\, dm=0$.

Theorems & Definitions (41)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Theorem 3
  • proof
  • Remark 4
  • Proposition 5
  • Theorem 6
  • proof
  • ...and 31 more