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Monotone increment processes, classical Markov processes, and Loewner chains

Volume 552 / 2020

Uwe Franz, Takahiro Hasebe, Sebastian Schleißinger Dissertationes Mathematicae 552 (2020), 1-119 MSC: Primary 30C35, 46L53, 60G51; Secondary 30C55, 30C80, 46L54, 46N30, 60E07, 60E10, 81R15, 81S25. DOI: 10.4064/dm808-1-2020 Published online: 31 August 2020


We prove one-to-one correspondences between certain decreasing Loewner chains in the upper half-plane, a special class of real-valued Markov processes, and quantum stochastic processes with monotonically independent additive increments. This leads us to a detailed investigation of probability measures on $\mathbb R$ with univalent Cauchy transform. We discuss several subclasses of such measures and obtain characterizations in terms of analytic and geometric properties of the corresponding Cauchy transforms.

Furthermore, we obtain analogous results for the setting of decreasing Loewner chains in the unit disk, which correspond to quantum stochastic processes of unitary operators with monotonically independent multiplicative increments.


  • Uwe FranzDépartement de mathématiques de Besançon Université de Bourgogne Franche-Comté
    16, route de Gray
    F-25 030
    Besançon Cedex, France
  • Takahiro HasebeDepartment of Mathematics
    Hokkaido University
    North 10, West 8, Kita-ku
    Sapporo 060-0810, Japan
  • Sebastian SchleißingerUniversity of Würzburg
    Emil-Fischer-Straße 40
    97074 Würzburg, Germany

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