2021/09/06 by Jedidi, Wissem, Jurek, Zbigniew J., Taymani, Nuha
#44A20 #60E07 #60E10 #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.2109.02709
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 (\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 ξ, the function t ↦ tn \er[ξt-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, ξ is the inverse time subordinator of a spectrally negative Lévy process, if, and only if, for some p≥ 1, the function t ↦ t \er[ξt-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.