Ona relation between classical and free infinitely divisible transforms

TL;DR

This paper explores the connection between classical and free infinitely divisible measures by identifying their Voiculescu transforms, deriving formulas for free-selfdecomposable measures, and illustrating methods with hyperbolic functions.

Contribution

It introduces a new approach to find free-probability analogues of classical measures by identifying their transforms and deriving differential equations for free-selfdecomposable measures.

Findings

Derived formulas for background driving transforms of free-selfdecomposable measures

Identified Voiculescu transforms linking classical and free measures

Provided examples using hyperbolic characteristic functions

Abstract

We study two ways (levels) of finding free-probability analogues of classical infinitely divisible measures. More precisely, we identify their Voiculescu transforms. For free-selfdecomposable measures we found the formula (a differential equation) for their background driving transforms. We illustrate our methods on the hyperbolic characteristic functions. As a by-product our approach may produce new formulas for some definite integrals.