Additive processes on the real line and Loewner chains

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 Loewner chains, a special kind of family of conformal mappings, on the upper half-plane. Generalizing the celebrated Loewner differential equation, we formulate an integral equation and the concept of ``generator'' for any Loewner chain of reciprocal Cauchy transforms. This generalization enables us to remove the assumption of the absolute continuity of Loewner chains which had been imposed in the literature. 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 Loewner chains, the set of monotone convolution hemigroups, and the set of classical convolution hemigroups on the real line. We also discuss similar homeomorphisms to free, boolean and anti-monotone convolution hemigroups on the real line.