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

We establish a new connection between the class of Nevanlinna-Pick functions and the one of the exponents associated to spectrally negative L\'evy 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 ({\it On subordinators, self-similar 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 $\xi$, the function $t \mapsto t^n \, \er[\xi_t^{-p}]$ 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, $\xi$ is the inverse time subordinator of a spectrally negative L\'evy process, if, and only if, for some $\,p\geq 1$, the function $t \mapsto t \, \er[\xi_t^{-p}]$ 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.