MULTIPLICATIVE STOCHASTIC PERTURBATIONS OF ONE-DIMENSIONAL MAPS

Abstract

We consider random perturbations of some one-dimensional map S : [0, 1] → [0, 1] such that [Formula: see text] parametrized by 0 < ε < 1, where {Cn} is an i.i.d. sequence. We prove that this random perturbation is small with respect to the noise level 0 < ε < 1 and give a class of one-dimensional maps for which there always exists a smooth invariant probability measure for the Markov process {Xn}n≥0.