  We consider random perturbations of a topologically transitive local
diffeomorphism of a Riemannian manifold. We show that if an absolutely
continuous ergodic stationary measures is expanding (all Lyapunov exponents
positive), then there is a random Gibbs-Markov-Young structure which can be
used to lift that measure. We also prove that if the original map admits a
finite number of expanding invariant measures then the stationary measures of a
sufficiently small stochastic perturbation are expanding.
