Generalized Many-Dimensional Excited Random Walk in Bernoulli Environment

TL;DR

This paper extends the generalized excited random walk to include a random Bernoulli environment, proving ballisticity for all probabilities p and establishing a Law of Large Numbers and Central Limit Theorem under certain conditions.

Contribution

It introduces a new model of excited random walk in a Bernoulli environment and proves its ballisticity, Law of Large Numbers, and Central Limit Theorem.

Findings

Proves ballisticity for all p in (0,1]

Establishes Law of Large Numbers under i.i.d. regeneration times

Derives Central Limit Theorem under stronger assumptions

Abstract

We study an extension of the generalized excited random walk (GERW) on $\mathbb{Z}^d$ introduced in [Ann. Probab. 40 (5), 2012, [7]] by Menshikov, Popov, Ram\'irez and Vachkovskaia. Our extension consists in studying a version of the GERW where excitation depends on a random environment. Given $p \in (0,1]$ (a parameter of the model) whenever the process visits a site for the first time, with probability $p$ it gains a drift in a given direction (could be any direction of the unit sphere). Otherwise, with probability $1-p$, it behaves as a $d$-martingale with zero-mean vector. Whenever the process visits an already-visited site, the process acts again as a $d$-martingale with zero-mean vector. We refer to the model as a GERW in Bernoulli environment, in short $p$-GERW. Under the same hypothesis of [7] (bounded jumps, uniform ellipticity), we show that the $p$-GERW is ballistic for all $p\in (0,1]$. Under the stronger assumptions that the increments of the regeneration times associated to the $p$-GERW are i.i.d. (condition which is satisfied, for example, for the excited random walk in a Bernoulli i.i.d. environment), we also obtain a Law of Large Numbers and a Central Limit Theorem.