Invariant Measures of Certain Markov Operators Describing a Homogeneous Random Medium

We are given a lamp in a row of lamps stretching to infinity on either side; if the lamp and an immediate neighbor are burning at a given instant, it will continue to burn at the next instant; otherwise, it will burn only with probability $\theta$. We show in this paper that, discounting the trivial case of “all lamps burning”, there is not more than one invariant measure on the space of states of this random medium.