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 [AA03], 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 L 1 -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 [ABV00] and the second one is the class of Viana maps.

In this paper we generalize [3] and prove that the class of accessible and saddle-conservative cocycles (a wide class which includes cocycles evolving in GLpd, Rq, SLpd, Rq and Sppd, Rq) L p -densely have a simple spectrum. We also generalize [3,1] and prove that for an L p -residual subset of accessible cocycles we have a one-point spectrum, by using a different approach of the one given in [3]. Finally, we show that the linear differential system versions of previous results also hold and give some applications.Remark 2.1. It follows from the definition of the metric and from Hölder inequality (see e.g.[37]) that, for all A, B P G and 1 ď p ď q ď 8, we have d p pA, Bq ď d q pA, Bq.2.1.3. Families of cocycles. We are interested in classes of maps A taking values in specific subgroups of GLpd, Rq. In the greater generality we consider subgroups that satisfy an accessability type condition.Example 1: The subgroups GLpd, Rq, SLpd, Rq, Spp2q, Rq, as well GLpd, Cq and SLpd, Cq, are accessible. Remark 2.2. In [13, Definition 1.2] the authors introduced a slightly different notion of accessability. See [13, Lemma 5.12] for a relation between those concepts.Next result shows that accessibility allows us to reach anywhere within the projective space and acting on elements of the group. Lemma 2.1. Let S be an accessible subgroup of GLpd, Rq. There exists K ą 0 such that, forProof. Fix some ǫ ą 0 and let 0 ă δ ă ǫ be such that if R 1 , R 2 P U δ :" tR P S : }R} ă δu, then R 2 R´1 1 P U ǫ . The hypothesis over S imply that for any w P RP d´1 , the evaluation map w : S Ñ RP d´1 given by A Þ Ñ Apwq is open, so that U δ pwq :" tRw : R P U δ u is an open subset of RP d´1 . Due to the compactness of the projective space one can writefor some m ě 1. Let u, v P RP d´1 be given with u P U δ pw i q and v P U δ pw j q for some i, j P t1, . . . , mu. Let R u , R v P U δ be such that R u w i " u, R v w j " v. There exist 1 ď k ď m and tR i u iďk , with }R i } ă ǫ, such that R k¨¨¨R1 w i " w j . Then R u,v u :" R v R k¨¨¨R1 R´1 u u " v and }R u,v } ď pm`2qǫ. We just have to consider K " pm`2qǫ.

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We consider random perturbations of non-uniformly expanding maps, possibly having a non-degenerate critical set. We prove that, if the Lebesgue measure of the set of points failing the non-uniform expansion or the slow recurrence to the critical set at a certain time, for almost all random orbits, decays in a (stretched) exponential fashion, then the decay of correlations along random orbits is stretched exponential, up to some waiting time. As applications, we obtain almost sure stretched exponential decay of random correlations for Viana maps, as for a class of non-uniformly expanding local diffeomorphisms and a quadratic family of interval maps.