Quantitative recurrence in two-dimensional extended processes

TL;DR

This paper investigates the recurrence properties of two-dimensional extended processes, including planar random walks and hyperbolic dynamical systems, establishing a relation between recurrence rates, process dimension, and return time distributions.

Contribution

It introduces a pointwise recurrence rate for extended processes and links it to the process dimension, providing new insights into their recurrence behavior.

Findings

Recurrence rate is related to the process dimension.

Rescaled return times converge in distribution.

Results apply to planar random walks and hyperbolic extensions.

Abstract

Under some mild condition, a random walk in the plane is recurrent. In particular each trajectory is dense, and a natural question is how much time one needs to approach a given small neighborhood of the origin. We address this question in the case of some extended dynamical systems similar to planar random walks, including $\ZZ^2$-extension of hyperbolic dynamics. We define a pointwise recurrence rate and relate it to the dimension of the process, and establish a convergence in distribution of the rescaled return times near the origin.