Integral transforms related to Nevanlinna-Pick functions from an analytic, probabilistic and free-probability point of view

Abstract

We establish a new connection between the class of Nevanlinna-Pick functions and the one of the exponents associated to spectrally negative Lévy processes. As a consequence, we compute the characteristics related to some hyperbolic functions and we show a property of temporal complete monotonicity, similar to the one obtained via the Lamperti transformation by Bertoin & Yor (On subordinators, selfsimilar Markov processes and some factorizations of the exponential variable, Elect. Comm. in Probab., vol. 6, pp. 95-106, 2001) for self-similar Markov processes. More precisely, we show the remarkable fact that for a subordinator ξ , the function t ↦ t n E[ξ −p t ] is , depending on the values of the exponents n = 0,1,2, p > −1, or a Bernstein function or a completely monotone function. In particular, ξ is the inverse time subordinator of a spectrally negative Lévy process, if, and only if, for some p ≥ 1, the function t ↦ t E[ξ −p t ] is a Stieltjes transform. Finally, we clarify to which extent Nevanlinna-Pick functions are related to free-probability and to Voiculescu transforms, and we provide an inversion procedure.