Limit theorems for self-intersecting trajectories in Z-extensions

TL;DR

This paper establishes a functional limit theorem for the self-intersection counts of trajectories in Z-extensions of dynamical systems, including the Lorentz gas, showing convergence to Brownian local time.

Contribution

It introduces a new limit theorem for self-intersecting trajectories in Z-extensions, linking discrete dynamical processes to continuous Brownian motion.

Findings

Proves a functional limit theorem for self-intersection processes.

Shows convergence of discrete local time to Brownian local time.

Applies results to Z-periodic Lorentz gas.

Abstract

This paper studies for a class of Z-extensions of dynamical systems including Z-periodic Lorentz gas the asymptotic behavior of the number of self-intersections of the trajectory of the flow. It concludes on a functional limit theorem for the auto-intersection process by studying Skorohod convergence of discrete local time of a dynamical random walk toward the local time of a Brownian motion.