Regularizing mappings of Lévy measures

In this paper we introduce and study a regularizing one-to-one mapping from the class of one-dimensional Lévy measures into itself. This mapping appeared implicitly in our previous paper [O.E. Barndorff-Nielsen, S. Thorbjørnsen, A connection between free and classical infinite divisibility, Inf. Dim. Anal. Quant. Probab. 7 (2004) 573-590], where we introduced a one-to-one mapping [Upsilon] from the class of one-dimensional infinitely divisible probability measures into itself. Based on the investigation of in the present paper, we deduce further properties of [Upsilon] . In particular it is proved that [Upsilon] maps the class of selfdecomposable laws onto the so called Thorin class . Further, partly motivated by our previous studies of infinite divisibility in free probability, we introduce a one-parameter family of one-to-one mappings , which interpolates smoothly between [Upsilon] ([alpha]=0) and the identity mapping on ([alpha]=1). We prove that each of the mappings shares many of the properties of [Upsilon]. In particular, they are representable in terms of stochastic integrals with respect to associated Lévy processes.