  We study random perturbations of multidimensional piecewise expanding maps.
We characterize absolutely continuous stationary measures (acsm) of randomly
perturbed dynamical systems in terms of pseudo-orbits linking the ergodic
components of absolutely invariant measures (acim) of the unperturbed system.
We focus on those components, called least-elements, which attract
pseudo-orbits. We show that each least element admits a neighbourhood which
supports exactly one ergodic acsm of the random system. We use this result to
identify random perturbations that exhibit a metastable behavior.
