There has recently been an increasing interest in the study of free and classical infinitely divisible distributions with Lévy measures involving fractional integrals.  The purpose of the talk is to present a review of the role played by the arcsine and other fractional measures in the construction of infinitely divisible distributions. Then we discuss a general framework, the relation to Upsilon transformations and the existence of corresponding stochastic integrals representations. New classes of infinitely divisible distributions are presented and its relation with well known classes is highlighted. Examples of Lévy measures involving modified Bessel and Whittaker functions are also presented.
The talk is based on several joint works with O. Arizmendi, O.E. Barndorff-Nielsen, M. Maejima and K.I. Sato.