Monotone convolution semigroups

Tom 200 / 2010

Takahiro Hasebe Studia Mathematica 200 (2010), 175-199 MSC: Primary 46L53, 46L54; Secondary 34A25. DOI: 10.4064/sm200-2-5


We study how a property of a monotone convolution semigroup changes with respect to the time parameter. Especially we focus on “time-independent properties”: in the classical case, there are many properties of convolution semigroups (or Lévy processes) which are determined at an instant, and moreover, such properties are often characterized by the drift term and Lévy measure. In this paper we exhibit such properties of monotone convolution semigroups; an example is the concentration of the support of a probability measure on the positive real line. Most of them are characterized by the same conditions on drift terms and Lévy measures as known in probability theory. These kinds of properties are mapped bijectively by a monotone analogue of the Bercovici–Pata bijection. Finally we compare such properties with classical, free, and Boolean cases, which will be important in an approach to unify these notions of independence.


  • Takahiro HasebeGraduate School of Science
    Kyoto University
    Kyoto 606-8502, Japan

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