Strong stochastic stability for non-uniformly expanding maps

Abstract: We consider random perturbations of discrete-time dynamical systems. We give sufficient conditions for the stochastic stability of certain classes of maps, in a strong sense. This improves the main result in J. F. Alves, V. Araujo, Random perturbations of non-uniformly expanding maps, Asterisque 286 (2003), 25--62, where it was proved the convergence of the stationary measures of the random process to the SRB measure of the initial system in the weak* topology. Here, under slightly weaker assumptions on the random perturbations, we obtain a stronger version of stochastic stability: convergence of the densities of the stationary measures to the density of the SRB measure of the unperturbed system in the L1-norm. As an application of our results we obtain strong stochastic stability for two classes of non-uniformly expanding maps. The first one is an open class of local diffeomorphisms introduced in J. F. Alves, C. Bonatti, M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly expanding, Invent. Math. 140 (2000), 351--398, and the second one the class of Viana maps.