Random perturbations of transformations of an interval

Let μ^ε be invariant measures of the Markov chains x _n ^F which are small random perturbations of an endomorphism f of the interval [0,1] satisfying the conditions of Misiurewicz [6]. It is shown here that in the ergodic case μ^ε converges as ε→0 to the smooth f-invariant measure which exists according to [6]. This result exhibits the first example of stability with respect to random perturbations while stability with respect to deterministic perturbations does not take place.