Reconstructing the environment seen by a RWRE
Jan Nagel (Technische Universität München)
Abstract
Consider a walker performing a random walk in an i.i.d. random environment, and assume that the walker tells us at each time the environment it sees at its present location. Given this history of the transition probabilities seen from the walker - but not its trajectory - can we reconstruct the law of the environment? We show that in a one-dimensional environment, the law of the environment can be reconstructed. This model can be seen as a special case of a scenery reconstruction problem, where the steps of the random walker depend on the scenery.
Full Text: Download PDF | View PDF online (requires PDF plugin)
Pages: 1-9
Publication Date: May 7, 2014
DOI: 10.1214/ECP.v19-3013
References
- Adelman, Omer; Enriquez, Nathanaël. Random walks in random environment: what a single trajectory tells. Israel J. Math. 142 (2004), 205--220. MR2085716
- Benjamini, Itai; Kesten, Harry. Distinguishing sceneries by observing the scenery along a random walk path. J. Anal. Math. 69 (1996), 97--135. MR1428097
- den Hollander, Frank; Steif, Jeffrey E. Random walk in random scenery: a survey of some recent results. Dynamics & stochastics, 53--65, IMS Lecture Notes Monogr. Ser., 48, Inst. Math. Statist., Beachwood, OH, 2006. MR2306188
- Kesten, Harry. Distinguishing and reconstructing sceneries from observations along random walk paths. Microsurveys in discrete probability (Princeton, NJ, 1997), 75--83, DIMACS Ser. Discrete Math. Theoret. Comput. Sci., 41, Amer. Math. Soc., Providence, RI, 1998. MR1630410
- Matzinger, Heinrich. Reconstructing a three-color scenery by observing it along a simple random walk path. Random Structures Algorithms 15 (1999), no. 2, 196--207. MR1704344
- Matzinger, Heinrich. Reconstructing a two-color scenery by observing it along a simple random walk path. Ann. Appl. Probab. 15 (2005), no. 1B, 778--819. MR2114990
- Löwe, Matthias; Matzinger, Heinrich, III. Scenery reconstruction in two dimensions with many colors. Ann. Appl. Probab. 12 (2002), no. 4, 1322--1347. MR1936595
- Solomon, Fred. Random walks in a random environment. Ann. Probability 3 (1975), 1--31. MR0362503
- Zeitouni, Ofer. Random walks in random environment. Lectures on probability theory and statistics, 189--312, Lecture Notes in Math., 1837, Springer, Berlin, 2004. MR2071631
This work is licensed under a Creative Commons Attribution 3.0 License.