Regularizing mappings of Lévy measures

In this paper we introduce and study a regularizing one-to-one mapping ϒ0 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 ϒϒ from the class ID(*)ID(*) of one-dimensional infinitely divisible probability measures into itself. Based on the investigation of ϒ0 in the present paper, we deduce further properties of ϒϒ . In particular it is proved that ϒϒ maps the class L(*)L(*) of selfdecomposable laws onto the so called Thorin class T(*)T(*). Further, partly motivated by our previous studies of infinite divisibility in free probability, we introduce a one-parameter family (ϒα)α∈[0,1] of one-to-one mappings ϒα:ID(*)→ID(*), which interpolates smoothly between ϒϒ (α=0α=0) and the identity mapping on ID(*)ID(*) (α=1α=1). We prove that each of the mappings ϒα shares many of the properties of ϒϒ. In particular, they are representable in terms of stochastic integrals with respect to associated Levy processes.