Open Access
2021 Reconstructing a recurrent random environment from a single trajectory of a Random Walk in Random Environment with errors
Jonas Jalowy, Matthias Löwe
Author Affiliations +
Electron. Commun. Probab. 26: 1-12 (2021). DOI: 10.1214/21-ECP425


We consider one infinite path of a Random Walk in Random Environment (RWRE, for short) in an unknown environment. This environment consists of either i.i.d. site or bond randomness. At each position the random walker stops and tells us the environment it sees at the point where it is, without telling us, where it is. These observations χ are spoiled by reading errors that occur with probability p<1. We show: If the RWRE is recurrent and satisfies the standard assumptions on such RWREs, then with probability one in the environment, the errors, and the random walk we are able reconstruct the law of the environment. For most situations this result is even independent of the value of p. If the distribution of the environment has a non-atomic part, we can even reconstruct the environment itself, up to translation.

Funding Statement

Research was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy EXC 2044-390685587, Mathematics Münster: Dynamics-Geometry-Structure. JJ is funded by the DFG through the SPP 2265 Random Geometric Systems


We are grateful to Nina Gantert for many hints on the behavior of RWRE. We also thank an anonymous referee and and anonymous Associate Editor for many useful remarks that spotted a mistake in the first version and helped to improve the paper.


Download Citation

Jonas Jalowy. Matthias Löwe. "Reconstructing a recurrent random environment from a single trajectory of a Random Walk in Random Environment with errors." Electron. Commun. Probab. 26 1 - 12, 2021.


Received: 28 August 2020; Accepted: 30 August 2021; Published: 2021
First available in Project Euclid: 6 December 2021

Digital Object Identifier: 10.1214/21-ECP425

Primary: 60J10 , 60K37

Keywords: Random walk in random environment , Random walk in random scenery , reconstruction

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