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A local limit theorem for a random walk in an intermittent dynamical environment

Juho Leppänen

TL;DR

The paper studies a one-dimensional extended dynamical system with intermittent, non-uniformly expanding local maps and shows that the horizontal component of the induced random walk on ${ Z}_+$ satisfies a strong law of large numbers, a central limit theorem, and a non-Gaussian local limit theorem in deterministic environments. It then establishes quenched LLTs for random, polynomially mixing environments, under precise moment and correlation decay conditions, by relating first-entry/return times to Gaussian approximations and carefully controlling approximation errors. The LLT takes the form of a weighted sum of Gaussian densities, with weights determined by the environmental configuration, linking transport properties to environment structure through first-entry statistics. Overall, the work extends Leskelä–Stenlund’s framework to intermittent dynamics, providing a rigorous local limit description of transport in inhomogeneous intermittent media and contributing to the theory of random walks in dynamical environments.

Abstract

We study an extended dynamical system on the non-negative real line with piecewise linear non-uniformly expanding local dynamics. With a uniformly distributed initial state, the distribution of successive states coincides with that of a random walk in an inhomogeneous environment. Under suitable conditions on the environment, we establish a central limit theorem and a (non-Gaussian) local limit theorem for the walk. Our approach builds on the work of Leskelä and Stenlund (Stochastic Process. Appl. 121(12), 2011), who analyzed a corresponding model with uniformly expanding local dynamics.

A local limit theorem for a random walk in an intermittent dynamical environment

TL;DR

The paper studies a one-dimensional extended dynamical system with intermittent, non-uniformly expanding local maps and shows that the horizontal component of the induced random walk on satisfies a strong law of large numbers, a central limit theorem, and a non-Gaussian local limit theorem in deterministic environments. It then establishes quenched LLTs for random, polynomially mixing environments, under precise moment and correlation decay conditions, by relating first-entry/return times to Gaussian approximations and carefully controlling approximation errors. The LLT takes the form of a weighted sum of Gaussian densities, with weights determined by the environmental configuration, linking transport properties to environment structure through first-entry statistics. Overall, the work extends Leskelä–Stenlund’s framework to intermittent dynamics, providing a rigorous local limit description of transport in inhomogeneous intermittent media and contributing to the theory of random walks in dynamical environments.

Abstract

We study an extended dynamical system on the non-negative real line with piecewise linear non-uniformly expanding local dynamics. With a uniformly distributed initial state, the distribution of successive states coincides with that of a random walk in an inhomogeneous environment. Under suitable conditions on the environment, we establish a central limit theorem and a (non-Gaussian) local limit theorem for the walk. Our approach builds on the work of Leskelä and Stenlund (Stochastic Process. Appl. 121(12), 2011), who analyzed a corresponding model with uniformly expanding local dynamics.

Paper Structure

This paper contains 18 sections, 16 theorems, 200 equations, 1 figure.

Table of Contents

  1. Introduction
  2. A family of intermittent maps
  3. Model definition
  4. A piecewise linear intermittent extended dynamical system
  5. Random walk from intermittent dynamics
  6. Statement of main results
  7. Definitions
  8. Limit theorems for deterministic environments
  9. Quenched limit theorems for random environments
  10. Proofs
  11. Proof of Theorem \ref{['thm:slln']}
  12. Proof of Theorem \ref{['thm:clt']}
  13. Proof of Theorem \ref{['thm:llt']}
  14. Estimate on $E_1(x,n)$
  15. Estimate on $E_2(x,n)$
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $\alpha < 1/2$. Suppose that $\varphi$ cannot be written as $\varphi = c + \psi - \psi \circ T + \lambda q$ for $\lambda > 0$, $c \in {\mathbb R}$, and some measurable functions $q : [0,1] \to {\mathbb Z}$, $\psi : [0,1] \to {\mathbb R}$. Then, for any bounded interval $J \subset {\mathbb R}$, a Here, $\phi_{\mu, \sigma^2}$ denotes the density of $N(\mu, \sigma^2)$:

Figures (1)

  • Figure 1: Transition kernel ${\mathcal{K}}$.

Theorems & Definitions (35)

  • Theorem 1.1: Special case of Theorem 1.2 in G05
  • Proposition 1.2
  • proof
  • Proposition 1.3
  • proof
  • Theorem 2.1: Strong law of large numbers
  • Remark 2.2
  • Theorem 2.3: Central limit theorem
  • Remark 2.4
  • Theorem 2.5: Local limit theorem
  • ...and 25 more