Random iteration and Markov operators

Assume that (Omega, A, P) is a probability space and X is a metric space. Given a product measurable function f : X x Omega -> X, we examine connections between the iterates f(n) : X x Omega(N) -> X (in the sense of K. Baron and M. Kuczma, Colloq. Math. 37 (1977), 263-269) of f and the Markov operator P with adjoint P* of the form P* phi(x) = integral(Omega)phi(f (x,omega)) P(d omega). Moreover, some results concerning the existence and the uniqueness of solutions phi : X -> R of the equation P* phi = phi will be also presented.