Additive processes on the real line and Loewner chains

TL;DR

This paper explores the connection between additive processes, various types of convolution semigroups in non-commutative probability, and Loewner chains, introducing generators to analyze convergence and relationships among different convolution types.

Contribution

It introduces the concept of generators for Loewner chains and establishes homeomorphisms linking different convolution hemigroups and Loewner chains, broadening understanding of their interrelations.

Findings

Generators characterize convergence of Loewner chains.

Homeomorphisms connect Loewner chains with convolution hemigroups.

Results apply to classical, free, and boolean convolutions without moment restrictions.

Abstract

This paper investigates additive processes with respect to several different independences in non-commutative probability in terms of the convolution hemigroups of the distributions of the increments of the processes. In particular, we focus on the relation of monotone convolution hemigroups and chordal Loewner chains, a special kind of family of conformal mappings. Generalizing the celebrated Loewner differential equation, we introduce the concept of ``generator'' for a class of Loewner chains. The locally uniform convergence of Loewner chains is then equivalent to a suitable convergence of generators. Using generators, we define homeomorphisms between the aforementioned class of chordal Loewner chains, the set of monotone convolution hemigroups with finite second moment, and the set of classical convolution hemigroups on the real line with finite second moment. Moreover, we define similar homeomorphisms between classical, free, and boolean convolution hemigroups on the real line, but without any assumptions on the moments.