Stiffness of random walks with reflecting boundary conditions

TL;DR

This paper investigates how the distribution of occupation times for a one-dimensional random walk with reflecting boundaries evolves from bimodal to unimodal over time, using spectral analysis and simulations.

Contribution

It provides a detailed spectral analysis of occupation time distributions for reflected random walks, connecting short-term bimodal and long-term unimodal behaviors.

Findings

Short-time distribution matches classical bimodal form.

Long-time distribution converges to a delta function.

Spectral analysis aligns well with numerical simulations.

Abstract

We study the distribution of occupation times for a one-dimensional random walk restricted to a finite interval by reflecting boundary conditions. At short times the classical bimodal distribution due to L\'evy is reproduced with walkers staying mostly either left or right to the initial point. With increasing time, however, the boundaries suppress large excursions from the starting point, and the distribution becomes unimodal converging to a $\delta$-distribution in the long time limit. An approximate spectral analysis of the underlying Fokker-Planck equation yields results in excellent agreement with numerical simulations.