Slow-fast systems in infinite measure, with or without averaging

TL;DR

This paper investigates the long-term behavior of differential equations influenced by fast flows with infinite measure, revealing different phenomena depending on the nature of the perturbation, with applications to Lorentz gas and geodesic flows.

Contribution

It introduces a comprehensive analysis of slow-fast systems in infinite measure settings, distinguishing between averaging and non-averaging regimes, and establishes new limit theorems for these cases.

Findings

Identifies two distinct asymptotic behaviors based on perturbation type.

Establishes limit theorems for non-stationary Birkhoff integrals in infinite measure.

Applies results to Lorentz gas and geodesic flow models.

Abstract

This paper studies the asymptotic behaviour of the solution of a differential equation perturbed by a fast flow preserving an infinite measure. This question is related with limit theorems for non-stationary Birkhoff integrals. We distinguish two settings with different behaviour: the integrable setting (no averaging phenomenon) and the case of an additive "centered" perturbation term (averaging phenomenon). The paper is motivated by the case where the perturbation comes from the Z-periodic Lorentz gas flow or from the geodesic flow over a Z-cover of a negatively curved compact surface. We establish limit theorems in more general contexts.