Almost sure invariance principle for random dynamical systems via Gou\"ezel 's approach

TL;DR

This paper extends Gou"ezel's spectral method to establish an almost sure invariance principle for non-stationary and random dynamical systems, including Anosov diffeomorphisms and billiard maps, under weaker covariance control.

Contribution

It introduces a generalized spectral approach for the vector-valued ASIP applicable to non-stationary sequences with linear covariance growth, and applies it to various random dynamical systems.

Findings

Proves quenched vector-valued ASIP for random perturbations of Anosov diffeomorphisms.

Establishes ASIP for random billiard maps in the Lorentz gas.

Extends Gou"ezel's method to systems with weaker covariance control.

Abstract

We extend the spectral approach of S. Gou\"ezel for the vector-valued almost sure invariance principle (ASIP) to certain classes of non-stationary sequences with a weaker control over the behavior of the covariance matrices, assuming only linear growth. Then we apply this extension to obtain a quenched vector-valued ASIP for random perturbations of a fixed Anosov diffeomorphism as well as random perturbations of a billiard map associated to the periodic Lorentz gas. We also consider certain classes of random piecewise expanding maps.