Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-19T12:59:31.772Z Has data issue: false hasContentIssue false
  • English
  • Français

A comparison of random walks in dependent random environments

Part of: Special processes Equilibrium statistical mechanics Stochastic processes

Published online by Cambridge University Press:  24 March 2016

Werner R. W. Scheinhardt  and
Dirk P. Kroese
Werner R. W. Scheinhardt*
Affiliation:
University of Twente
Dirk P. Kroese*
Affiliation:
The University of Queensland
*
* Postal address: Department of Applied Mathematics, University of Twente, PO Box 217, 7500 AE Enschede, The Netherlands. Email address: w.r.w.scheinhardt@utwente.nl
** Postal address: School of Mathematics and Physics, The University of Queensland, St Lucia, Brisbane, 4072, Australia. Email address: kroese@maths.uq.edu.au
Get access Rights & Permissions [Opens in a new window]

Abstract

We provide exact computations for the drift of random walks in dependent random environments, including k-dependent and moving average environments. We show how the drift can be characterized and evaluated using Perron–Frobenius theory. Comparing random walks in various dependent environments, we demonstrate that their drifts can exhibit interesting behavior that depends significantly on the dependency structure of the random environment.

Keywords

Random walkdependent random environmentdriftPerron–Frobenius eigenvalue

MSC classification

Primary: 60K37: Processes in random environments 60G50: Sums of independent random variables; random walks
Secondary: 82B41: Random walks, random surfaces, lattice animals, etc.
Type
Research Article
Information
Advances in Applied Probability , Volume 48 , Issue 1 , March 2016 , pp. 199 - 214
Copyright
Copyright © Applied Probability Trust 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Alili, S. (1999). Asymptotic behaviour for random walks in random environments. J. Appl. Prob. 36, 334–349. CrossRefGoogle Scholar
[2]Bean, N. G.et al. (1997). The quasi-stationary behavior of quasi-birth-and-death processes. Ann. Appl. Prob. 7, 134–155. Google Scholar
[3]Brereton, T.et al. (2012). Efficient simulation of charge transport in deep-trap media. In Proc. 2012 Winter Simulation Conference (Berlin), IEEE, New York, pp. 1–12. Google Scholar
[4]Chernov, A. A. (1962). Replication of multicomponent chain by the 'lighting mechanism'. Biophysics 12, 336–341. Google Scholar
[5]Dolgopyat, D., Keller, G. and Liverani, C. (2008). Random walk in Markovian environment. Ann. Prob. 36, 1676–1710. CrossRefGoogle Scholar
[6]Greven, A. and den Hollander, F. (1994). Large deviations for a random walk in random environment. Ann. Prob. 22, 1381–1428. CrossRefGoogle Scholar
[7]Hughes, B. D. (1996). Random Walks and Random Environments, Vol. 2. Oxford University Press. Google Scholar
[8]Kesten, H., Kozlov, M. W. and Spitzer, F. (1975). A limit law for random walk in a random environment. Compositio Math. 30, 145–168. Google Scholar
[9]Kozlov, S. M. (1985). The method of averaging and walks in inhomogeneous enviroments. Russian Math. Surveys 40, 73–145. CrossRefGoogle Scholar
[10]Mayer-Wolf, E., Roitershtein, A. and Zeitouni, O. (2004). Limit theorems for one-dimensional transient random walks in Markov environments. Ann. Inst. H. Poincaré Prob. Statist. 40, 635–659. Google Scholar
[11]Révész, P. (2013). Random Walk in Random and Non-Random Environments, 3rd edn. World Scientific, Hackensack, NJ. CrossRefGoogle Scholar
[12]Scheinhardt, W. R. W. and Kroese, D. P. (2014). Computing the drift of random walks in dependent random environments. Preprint. Available at http://arxiv.org/abs/1406.3390v1. Google Scholar
[13]Sinai, Y. G. (1983). The limiting behavior of a one-dimensional random walk in a random medium. Theory Prob. Appl. 27, 256–268. Google Scholar
[14]Solomon, F. (1975). Random walks in a random environment. Ann. Prob. 3, 1–31. CrossRefGoogle Scholar
[15]Stenzel, O.et al. (2014). A general framework for consistent estimation of charge transport properties via random walks in random environments. Multiscale Model. Simul. 12, 1108–1134. Google Scholar
[16]Sznitman, A.-S. (2004). Topics in random walks in random environment. In School and Conference on Probability Theory (ICTP Lecture Notes XVII), Abdus Salem, Trieste, pp. 203–266. Google Scholar
[17]Temkin, D. E. (1969). The theory of diffusionless crystal growth. J. Crystal Growth 5, 193–202. CrossRefGoogle Scholar
[18]Zeitouni, O. (2004). Part II: Random walks in random environment. In Lectures on Probability Theory and Statistics (Lecture Notes Math. 1837), Springer, Berlin, pp. 189–312. CrossRefGoogle Scholar
[19]Zeitouni, O. (2012). Random walks in random environment. In Computational Complexity, Springer, New York, pp. 2564–2577. CrossRefGoogle Scholar