Reconstructing the environment seen by a RWRE

TL;DR

This paper investigates whether the environment of a one-dimensional random walk in a random environment (RWRE) can be reconstructed from the observed transition probabilities, determining recurrence or transience without knowing the walker's exact path.

Contribution

It demonstrates that in a one-dimensional setting, the environment's law can be reconstructed solely from the observed environment history, including the recurrence or transience status.

Findings

The environment law can be reconstructed from observed transition probabilities.

Recurrence or transience of the RWRE can be determined from environment observations.

Reconstruction is possible without access to the walker's exact trajectory.

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 tell if the RWRE is recurrent or transient? Can we reconstruct the law of the environment? We show that in a one-dimensional environment, the law of the environment can be reconstructed, and we know in particular if the RWRE is recurrent or transient.