Consider a random walk in an irreducible random environment on the positive integers. We prove that the annealed law of the random walk determines uniquely the law of the random environment. An application to linearly edge-reinforced random walk is given.
"Uniqueness of the mixing measure for a random walk in a random environment on the positive integers." Electron. Commun. Probab. 14 31 - 35, 2009. https://doi.org/10.1214/ECP.v14-1441