## Institute of Mathematical Statistics Lecture Notes - Monograph Series

- Lecture Notes--Monograph Series
- Volume 48, 2006, 53-65

### Random walk in random scenery: A survey of some recent results

Jeffrey E. Steif and Frank den Hollander

#### Abstract

In this paper we give a survey of some recent results for random walk in random scenery (RWRS). On $\Z^d$, $d\geq 1$, we are given a random walk with i.i.d. increments and a random scenery with i.i.d. components. The walk and the scenery are assumed to be independent. RWRS is the random process where time is indexed by $\Z$, and at each unit of time both the step taken by the walk and the scenery value at the site that is visited are registered. We collect various results that classify the ergodic behavior of RWRS in terms of the characteristics of the underlying random walk (and discuss extensions to stationary walk increments and stationary scenery components as well). We describe a number of results for scenery reconstruction and close by listing some open questions.

#### Chapter information

**Source***Dynamics & Stochastics: Festschrift in honor of M. S. Keane* (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2006)

**Dates**

First available in Project Euclid: 28 November 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.lnms/1196285808

**Digital Object Identifier**

doi:10.1214/074921706000000077

**Mathematical Reviews number (MathSciNet)**

MR2306188

**Zentralblatt MATH identifier**

1124.60084

**Subjects**

Primary: 60G10: Stationary processes

Secondary: 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs

**Keywords**

random walk in random scenery Bernoulli K-automorphism weak Bernoulli finitary coding conditional probability distribution bad configuration scenery reconstruction

**Rights**

Copyright © 2006, Institute of Mathematical Statistics

#### Citation

den Hollander, Frank; Steif, Jeffrey E. Random walk in random scenery: A survey of some recent results. Dynamics & Stochastics, 53--65, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2006. doi:10.1214/074921706000000077. https://projecteuclid.org/euclid.lnms/1196285808