Invariant measures for random dynamical systems on the circle

Abstract.

We show that ergodic invariant measures of random dynamical systems induced by stochastic differential equations on the circle S ¹ can only be either fixed, in which case they either have to be a deterministic fixed point, or have full support, or they can be random point measures, measurable either with respect to the past or with respect to the future. Associated with an ergodic invariant point measure, measurable with respect to the past, there is an ergodic invariant point measure, measurable with respect to the future, which has the same number of points. Furthermore, between any two of the points supporting one of these measures there is a point from the support of the other one.