Simple random walk on Z^2 perturbed on the axis (renewal case)

TL;DR

This paper analyzes a two-dimensional simple random walk influenced by a force on the axis, revealing a renewal structure that leads to ergodic properties of the walk's segments on the axis, with implications for physical particle models.

Contribution

It introduces a parametrization of the force on the axis that induces a renewal structure in the walk, providing new insights into the walk's long-term behavior.

Findings

Existence of a renewal structure in the perturbed walk

Ergodic properties for trajectory segments on the axis

Parametrization of force leading to renewal behavior

Abstract

We study a simple random walk on Z^2 with constraints on the axis. Motivation comes from physics when particles (a gas for example, see [Dal88]) are submitted to a local field. In our case we assume that the particle evolves freely in the cones but when touching the axis a force pushes it back progressively to the origin. The main result proves that this force can be parametrized in such a way that a renewal structure appears in the trajectory of the random walk. This implies the existence of an ergodic result for the parts of the trajectory restricted to the axis.