Scaling limits of recurrent excited random walks on integers

TL;DR

This paper investigates the scaling limits of recurrent excited random walks on integers, revealing their convergence to perturbed Brownian motions or maximum processes depending on the drift parameter.

Contribution

It characterizes the scaling limits of ERWs in i.i.d. cookie environments with bounded cookies, including boundary cases with logarithmic adjustments.

Findings

ERW converges to a perturbed Brownian motion when |delta|<1

At boundary |delta|=1, ERW converges to a multiple of the Brownian maximum

The results apply to environments with both positive and negative excitations

Abstract

We describe scaling limits of recurrent excited random walks (ERWs) on integers in i.i.d. cookie environments with a bounded number of cookies per site. We allow both positive and negative excitations. It is known that ERW is recurrent if and only if the expected total drift per site, delta, belongs to the interval [-1,1]. We show that if |delta|<1 then the diffusively scaled ERW under the averaged measure converges to a (delta,-delta)-perturbed Brownian motion. In the boundary case, |delta|=1, the space scaling has to be adjusted by an extra logarithmic term, and the weak limit of ERW happens to be a constant multiple of the running maximum of the standard Brownian motion, a transient process.