Monotone convolution semigroups

Takahiro Hasebe ¹

¹ Graduate School of Science Kyoto University Kyoto 606-8502, Japan

Studia Mathematica, Tome 200 (2010) no. 2, pp. 175-199

Résumé

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.

Détail

DOI : 10.4064/sm200-2-5

Keywords: study property monotone convolution semigroup changes respect time parameter especially focus time independent properties classical there many properties convolution semigroups processes which determined instant moreover properties often characterized drift term measure paper exhibit properties monotone convolution semigroups example concentration support probability measure positive real line characterized conditions drift terms measures known probability theory these kinds properties mapped bijectively monotone analogue bercovici pata bijection finally compare properties classical boolean cases which important approach unify these notions independence