Central limit theorems for the $\mathbb{Z}^2$-periodic Lorentz gas

TL;DR

This paper establishes central limit theorems for Birkhoff sums in $bZ^2$-extensions of dynamical systems, with applications to the $bZ^2$-periodic Lorentz process, demonstrating stochastic behavior and convergence to Gaussian limits.

Contribution

It proves central limit theorems for $bZ^2$-extensions of dynamical systems with spectral properties, specifically applying to the $bZ^2$-periodic Lorentz process.

Findings

Proved CLT for Birkhoff sums in $bZ^2$-extensions.

Established a functional CLT for the Lorentz process.

Applied results to both discrete and continuous time observables.

Abstract

This paper is devoted to the study of the stochastic properties of dynamical systems preserving an infinite measure. More precisely we prove central limit theorems for Birkhoff sums of observables of $\mathbb{Z}^2$-extensions of dynamical systems (satisfying some nice spectral properties). The motivation of our paper is the $\mathbb{Z}^2$-periodic Lorentz process for which we establish a functional central limit theorem for H\"older continuous observables (in discrete time as well as in continuous time).