Small stochastic perturbations of random maps with position dependent probabilities

A position dependent random map is a dynamical system consisting of a collection of maps Such that, at each iteration, a selection of a map is made randomly by means of probabilities which are functions of position. Let f* be an invariant density of the position dependent random map T-. We consider a model of small random perturbations of the random map T-epsilon For each epsilon > 0, Tepsilon has an invariant density function f(epsilon). We prove that fepsilon --> f* as epsilon --> 0.