On the liftability of expanding stationary measures

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.