Regularizing mappings of Levy measures

In this paper we introduce and study a regularizing one-to-one mapping Y-o from the class of one-dimensional Levy measures into itself. This mapping appeared implicitly in our previous paper [O.E. Barndorff-Nielsen, S. Thorbjornsen, 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 Yfrom the class of one-dimensional infinitely divisible probability measures into itself. Based on the investigation of Yo in the present paper, we deduce further properties of Y. In particular it is proved that Y maps the class L(*) of selfdecomposable laws onto the so called Thorin class T(*). Further, partly motivated by our previous studies of infinite divisibility in free probability, we introduce a one-parameter family (T-alpha)(alpha is an element of[0,1]) of one-to-one mappings Y-alpha:J D(*) --> J D(*), which interpolates smoothly between Y(alpha = 0) and the identity mapping on J D(*) (alpha = 1). We prove that each of the mappings Y-alpha shares many of the properties of Y. In particular, they are representable in terms of stochastic integrals with respect to associated Levy processes. (C) 2005 Elsevier B.V. All rights reserved.