Reconstructing a (recurrent) random environment from a single trajectory of Random Walk in Random Environment with errors

Matthias Löwe

Published 2020-08-27Version 1

We consider one infinite path of Random Walk in Random Environment (RWRE, for short) in an unknown i.i.d.\ environment $\omega$. At each position the random walker stops and tells us the environment it sees, without telling us, where it is. These observations $\chi$ are spoiled by reading errors. We show: If the error probability $p$ is smaller than 1, RWRE is recurrent (whenever the distribution of the environment has atomic parts), and the atomic parts of the distribution of the environment and the noise have disjoint support, with probability one in the environment and in the random walk we are able reconstruct the law of the environment: If the the law of the environment is not purely atomic we are even able to reconstruct the environment itself, up to translation.