INVARIANT DENSITIES FOR POSITION-DEPENDENT RANDOM MAPS ON THE REAL LINE: EXISTENCE, APPROXIMATION AND ERROR BOUNDS

Abstract

A random map is a discrete-time dynamical system in which a transformation is randomly selected from a collection of transformations according to a probability function and applied to the process. In this note, we study random maps with position-dependent probabilities on ℝ. This means that the random map under consideration consists of transformations which are piecewise monotonic with countable number of branches from ℝ into itself and a probability function which is position dependent. We prove existence of absolutely continuous invariant probability measures and construct a method for approximating their densities. Explicit quantitative bound on the approximation error is given.