Additive Processes on the Unit Circle and Loewner Chains

Abstract

Abstract This paper defines the notion of generators for a class of decreasing radial Loewner chains that 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 Lévy–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.