Additive processes on the unit circle and Loewner chains

TL;DR

This paper introduces generators for decreasing radial Loewner chains on the unit circle, generalizing Loewner's equation, and explores their connections with additive processes, non-commutative probability, and convolution semigroups.

Contribution

It defines a new class of generators for Loewner chains, linking them to additive processes, non-commutative probability, and convolution semigroups, extending classical theory.

Findings

Established a homeomorphism between Loewner chains and additive process distributions.

Connected generators with monotone and free convolution semigroups.

Computed generators for chains derived from free convolution hemigroups.

Abstract

This paper defines the notion of generators for a class of decreasing radial Loewner chains which are only continuous with respect to time. For this purpose, "Loewner's integral equation" which generalizes Loewner's differential equation is defined and analyzed. The definition of generators is motivated by the L\'evy-Khintchine representation for additive processes on the unit circle. Actually, we can and do introduce a homeomorphism between the above class of Loewner chains and the set of the distributions of increments of additive processes equipped with suitable topologies. On the other hand, from the viewpoint of non-commutative probability theory, the above generators also induce bijections with some other objects: in particular, monotone convolution hemigroups and free convolution hemigroups. Finally, the generators of Loewner chains constructed from free convolution hemigroups via subordination are computed.