RANDOM DYNAMICAL SYSTEMS ON ORDERED TOPOLOGICAL SPACES

Let (Xn, n ≥ 0) be a random dynamical system and its state space be endowed with a reasonable topology. Instead of completing the structure as common by some linearity, this study stresses – motivated in particular by economic applications – order aspects. If the underlying random transformations are supposed to be order-preserving, this results in a fairly complete theory. First of all, the classical notions of and familiar criteria for recurrence and transience can be extended from discrete Markov chain theory. The most important fact is provided by existence and uniqueness of a locally finite invariant measure for recurrent systems. It allows to derive ergodic theorems as well as to introduce an attractor in a natural way. The classification is completed by distinguishing positive and null recurrence corresponding, respectively, to the case of a finite or infinite invariant measure; equivalently, this amounts to finite or infinite mean passage times. For positive recurrent systems, moreover, strengthened versions of weak convergence as well as generalized laws of large numbers are available