Regularising Mappings of Lévy Measures
نویسندگان
چکیده
In this paper we introduce and study a regularising one-to-one mapping Υ0 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 Υ from the class ID(∗) of one-dimensional infinitely divisible probability measures into itself. Based on the studies of Υ0 in the present paper, we also 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(∗). Finally, partly motivated by our previous studies of infinite divisibility in free probability, we introduce a one-parameter family (Υ)α∈[0,1] of one-to-one mappings Υ : 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 shares the properties of Υ exhibited in [BT2]. In particular, they are representable in terms of stochastic integrals with respect to associated Levy processes.
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