Random walks and Lorentz processes
Domokos Szasz
TL;DR
This paper investigates whether local perturbations of planar stochastic systems—namely random walks and Lorentz ( Sinai ) processes—alter their limiting behavior and recurrence properties. By combining probabilistic limit theorems, local limit theorems, and ergodic/dynamical methods, the authors establish that planar finite-horizon perturbations preserve diffusive limits and recurrence, while infinite-horizon cases require non-standard scaling yet still exhibit Brownian limits with preserved covariance. The work extends Polya-type recurrence results from simple random walks to more complex planar dynamics, and analyzes both bounded and unbounded jump scenarios, highlighting open problems for strongly perturbed and unbounded-jump walks. The results contribute to a deeper understanding of robustness of limit laws under local perturbations and provide a framework for addressing Sinai’s 1981 question in broader settings.
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.