Random walks and Lorentz processes

TL;DR

This paper explores the probabilistic properties of random walks and Lorentz processes, demonstrating how local perturbations affect their limiting behaviors and recurrence, with new insights into longstanding open problems in dynamical systems and probability theory.

Contribution

It shows that local perturbations do not change limit laws for planar Lorentz processes and provides an alternative proof for their recurrence, addressing Sinai's 1981 question.

Findings

Local perturbations leave limit laws unchanged in planar Lorentz processes.

Probabilistic approach offers new insights into Lorentz process behavior.

Alternative proof of recurrence for finite horizon Lorentz processes.

Abstract

Random walks and Lorentz processes serve as fundamental models for Brownian motion. The study of random walks is a favorite object of probability theory, whereas that of Lorentz processes belongs to the theory of hyperbolic dynamical systems. Here, we first present examples where the method based on the probabilistic approach led to new insights into the study of the Lorentz process. Motivated by a 1981 question of Sinai about limiting laws for planar locally perturbed Lorentz processes, we first derived that, in the plane, local perturbations of homogeneous random walks leave the limit laws and the limiting processes unchanged - independently whether the walk had bounded or unbounded jumps. Afterward, we obtained probabilistic statements for local perturbations of planar Lorentz processes with finite horizon. (Similar statements for local perturbations of Lorentz processes with infinite horizon is a most interesting open problem!) Often an interesting consequence of a local limit theorem for a process is the recurrence of the same process. In this way, our approach also provides an alternative proof for a 2003 result of Lenci about the recurrence of the locally perturbed Lorentz processes with finite horizon. Afterward an unsolved problem, related to Sinai 1981 question, is formulated as an analogous problem in the language of random walks.