The Markov chain associated to a Pick function
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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 x2 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 L1(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.
KeywordsMarkov Chain Lebesgue Measure Integral Representation Compact Support Half Plane
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