Complex analysis methods in noncommutative probability

TL;DR

This thesis explores advanced convolution methods in noncommutative probability, establishing regularity results, connections between different convolutions, and properties of infinitely divisible measures within this mathematical framework.

Contribution

It introduces new regularity results for free convolutions, links Boolean and free convolutions, and characterizes infinitely divisible measures in noncommutative probability.

Findings

Proved regularity results for free convolutions

Established connections between Boolean and free convolutions

Characterized infinitely divisible measures in monotonic convolutions

Abstract

In this thesis we study convolutions that arise from noncommutative probability theory. We prove several regularity results for free convolutions, and for measures in partially defined one-parameter free convolution semigroups. We discuss connections between Boolean and free convolutions and, in the last chapter, we prove that any infinitely divisible probability measure with respect to monotonic additive or multiplicative convolution belongs to a one-parameter semigroup with respect to the corresponding convolution. Earlier versions of some of the results in this thesis have already been published, while some others have been submitted for publication. We have preserved almost entirely the specific format for PhD theses required by Indiana University. This adds several unnecessary pages to the document, but we wanted to preserve the specificity of the document as a PhD thesis at Indiana University.