MPS-RR 2004-26
November 2004

Regularising Mappings of Lévy Measures

Abstract

In this paper we introduce and study a regularising one-to-one mapping $\Upsilon_0$ from the class of one-dimensional Lévy measures into itself. This mapping appeared implicitly in our previous paper [BT2], where we introduced a one-to-one mapping $\Upsilon$ from the class $\mathcal{I}\mathcal{D}(*)$ of one-dimensional infinitely divisible probability measures into itself. Based on the studies of $\Upsilon_0$ in the present paper, we also deduce further properties of $\Upsilon$. In particular it is proved that $\Upsilon$ maps the class $\mathcal{L}(*)$ of selfdecomposable laws onto the so called Thorin class $\mathcal{T}(*)$. Finally, partly motivated by our previous studies of infinite divisibility in free probability, we introduce a one-parameter family $(\Upsilon^\alpha)_{\alpha \in [0,1]}$ of one-to-one mappings $\Upsilon^\alpha : \mathcal{I}\mathcal{D}(*) \to \mathcal{I}\mathcal{D}(*)$, which interpolates smoothly between $\Upsilon$ ($\alpha = 0$ ) and the identity mapping on $\mathcal{I}\mathcal{D}(*)$ ( $\alpha = 1$ ). We prove that each of the mappings $\Upsilon^\alpha$ shares the properties of $\Upsilon$ exhibited in [BT2]. In particular, they are representable in terms of stochastic integrals with respect to associated Lévy processes.

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