Electronic Journal of Probability

Excited random walk in a Markovian environment

Nicholas F. Travers

Full-text: Open access

Abstract

One dimensional excited random walk has been extensively studied for bounded, i.i.d. cookie environments. In this case, many important properties of the walk including transience or recurrence, positivity or non-positivity of the speed, and the limiting distribution of the position of the walker are all characterized by a single parameter $\delta $, the total expected drift per site. In the more general case of stationary ergodic environments, things are not so well understood. If all cookies are positive then the same threshold for transience vs. recurrence holds, even if the cookie stacks are unbounded. However, it is unknown if the threshold for transience vs. recurrence extends to the case when cookies may be negative (even for bounded stacks), and moreover there are simple counterexamples to show that the threshold for positivity of the speed does not. It is thus natural to study the behavior of the model in the case of Markovian environments, which are intermediate between the i.i.d. and stationary ergodic cases. We show here that many of the important results from the i.i.d. setting, including the thresholds for transience and positivity of the speed, as well as the limiting distribution of the position of the walker, extend to a large class of Markovian environments. No assumptions are made about the positivity of the cookies.

Article information

Source
Electron. J. Probab., Volume 23 (2018), paper no. 43, 60 pp.

Dates
Received: 17 May 2017
Accepted: 28 February 2018
First available in Project Euclid: 9 May 2018

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1525852820

Digital Object Identifier
doi:10.1214/18-EJP155

Mathematical Reviews number (MathSciNet)
MR3806411

Zentralblatt MATH identifier
1390.60358

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60K37: Processes in random environments 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Keywords
excited random walk cookie random walk self-interacting random walk Markovian environment random environment

Rights
Creative Commons Attribution 4.0 International License.

Citation

Travers, Nicholas F. Excited random walk in a Markovian environment. Electron. J. Probab. 23 (2018), paper no. 43, 60 pp. doi:10.1214/18-EJP155. https://projecteuclid.org/euclid.ejp/1525852820


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