open access

https://hdl.handle.net/2115/90845

Additive Processes on the Unit Circle and Loewner Chains

英語

2022-11

Oxford University Press

IMRN: International Mathematics Research Notices

2022(22)

17797-17848

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 Levy-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.

学術雑誌論文

Accepted Manuscript

This is a pre-copyedited, author-produced version of an article accepted for publication in IMRN: International Mathematics Research Notices following peer review. The version of record Volume 2022, Issue 22, November 2022, Pages 17797–17848 is available online at: https://academic.oup.com/imrn/article/2022/22/17797/6355422 and https://doi.org/10.1093/imrn/rnab157.

1073-7928