  We prove that the statistical properties of random perturbations of a
nonuniformly hyperbolic diffeomorphism are described by a finite number of
stationary measures. We also give necessary and sufficient conditions for the
stochastic stability of such dynamical systems. We show that a certain
$C^2$-open class of nonuniformly hyperbolic diffeomorphisms introduced in
[Alves, J; Bonatti, C. and Viana, V., SRB measures for partially hyperbolic
systems with mostly expanding central direction, Invent. Math., 140 (2000),
351-398] are stochastically stable. Our setting encompasses that of partially
hyperbolic diffeomorphisms as well. Moreover, the techniques used enable us to
obtain SRB measures in this setting through zero-noise limit measures.
