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Steen Thorbjørnsen

Regularizing mappings of Lévy measures

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  • Department of Mathematical Sciences
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 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 . In particular it is proved that 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 (α=0) and the identity mapping on (α=1). We prove that each of the mappings shares many of the properties of . In particular, they are representable in terms of stochastic integrals with respect to associated Levy processes.
Original languageEnglish
JournalStochastic Processes and Their Applications
Volume116
Issue3
Pages (from-to)423-446
Number of pages24
ISSN0304-4149
DOIs
Publication statusPublished - 2006

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