A random walk on Z with drift driven by its occupation time at zero

TL;DR

This paper studies a one-dimensional random walk with a drift influenced by its visits to zero, revealing different behaviors depending on the decay rate of the drift, including convergence to a symmetric exponential law.

Contribution

It introduces a novel model where the drift depends on occupation time at zero and characterizes its asymptotic regimes and limiting distributions.

Findings

For slow decay rates, the walk's position converges to a symmetric exponential distribution.

The walk's range scales differently from its position in certain regimes.

Distinct regimes are identified based on the drift decay rate.

Abstract

We consider a nearest neighbor random walk on the one-dimensional integer lattice with drift towards the origin determined by an asymptotically vanishing function of the number of visits to zero. We show the existence of distinct regimes according to the rate of decay of the drift. In particular, when the rate is sufficiently slow, the position of the random walk, properly normalized, converges to a symmetric exponential law. In this regime, in contrast to the classical case, the range of the walk scales differently from its position.