On a Method of Introducing Free-Infinitely Divisible Probability Measures

Abstract

Abstract. Random integral mappings Ih,r pa,bs give isomorphism between the subsemigroups of the classical pID,˚q and the free-infinite divisible pID, q probability measures. This allows us to introduce new examples of such measures, more precisely their corresponding characteristic functionals.

Ona relation between classical and free infinitely divisible transforms

“…The fact that the function given in (A), indeed, defines Voiculescu transform of an free-infinitely divisible measures was already shown in Jurek (2007), Corollary 6; cf. [12] (also repeated in [13]).…”

“…However, for the uniqueness questions, of the representation (3), is enough to consider Voiculescu transforms only on the imaginary axis; Jurek (2006) , cf. [11]; (also [12], [13] and [8]).…”

“…As in previous papers Jurek (2006), cf. [11] (or [12], [13]) we consider the transforms V ν only on the imaginary line.…”

“…[15]. The characteristic function ψ (in (13)) is referred to as the background driving characteristic function (BDCF) of φ ∈ L and Y ν the background driving Lévy process (BDLP) of µ. Remark 4.…”

Urbanik type subclasses of the free-infinitely divisible transforms

“…For the class (U <1> , ⊞) of free s-selfdecomposable measures we may use the identity Φ(−ix/t, 1, 2) = it(−x − it log(1 + ix/t)). The characterization of the class (U <1> ⊞) ≡ (U, ⊞) was earlier given in Jurek (2016), Proposition 1 (b). Note a misprint there: it should be t 2 ,no (it) 2 in part (b).…”