Regularizing mappings of Lévy measures

Abstract

In this paper we introduce and study a regularizing one-to-one mapping ϒ 0 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 ϒ from the class ID ( * ) of one-dimensional infinitely divisible probability measures into itself. Based on the investigation of ϒ 0 in the present paper, we deduce further properties of ϒ . In particular it is proved that ϒ 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 ( ϒ α ) α ∈ [ 0 , 1 ] of one-to-one mappings ϒ α : ID ( * ) → ID ( * ) , which interpolates smoothly between ϒ ( α = 0 ) and the identity mapping on ID ( * ) ( α = 1 ). We prove that each of the mappings ϒ α shares many of the properties of ϒ . In particular, they are representable in terms of stochastic integrals with respect to associated Levy processes.