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

Abstract

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 ${\mathit{\chi }}^{\prime }$ 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.