The Markov chain associated to a Pick function

To each function ϕ˜(ω) mapping the upper complex half plane ?⁺ into itself such that the coefficient of ω in the Nevanlinna integral representation is one, we associate the kernel p(y, dx) of a Markov chain on ℝ by The aim of this paper is to study this chain in terms of the measure μ appearing in the Nevanlinna representation of ϕ˜(ω). We prove in particular three results. If x ² is integrable by μ, a law of large numbers is available. If μ is singular, i.e. if ϕ˜ is an inner function, then the operator P on L ^∞(ℝ) for the Lebesgue measure is the adjoint of T defined on L ¹(ℝ) by T(f)(ω) = f(ϕ(ω)), where ϕ is the restriction of ϕ˜ to ℝ. Finally, if μ is both singular and with compact support, we give a necessary and sufficient condition for recurrence of the chain. Received: 24 April 1998 / Revised version: 13 March 2000 / Published online: 20 October 2000