Infinitely divisible central probability measures on compact Lie groups-regularity, semigroups and transition kernels
type: Item
instance of: scholarly article
title: Infinitely divisible central probability measures on compact Lie groups-regularity, semigroups and transition kernels
zbMATH Open document ID: 1237.60005
DOI: 10.1214/10-AOP604
author: David Applebaum
published in: The Annals of Probability
publication date: 2012-01-09T00:00:00Z
full work available at URL: https://arxiv.org/abs/1006.4711
review text: The investigation of central probabilities on locally compact abelian groups has a long history, with applications to probability theory as well as to analysis. E.g., the existence of central Gaussian laws has a strong impact on the underlying group, see e.g., the nice characterization of central groups by \textit{E. Siebert} [``Absolut-Stetigkeit und Träger von Gauß-Verteilungen auf lokalkompakten Gruppen'', Math. Ann. 210, 129--147 (1974; Zbl 0273.43003)]. Hence groups admitting a central Gaussian semigroup with full support are (connected) and central, in particular, MAP groups. The author considers central continuous convolution semigroups on \textit{compact} Lie groups, which behave similarly to the heat semigroup. More precisely, he induces a class \(\mathrm{CID}_{\mathbb{R}}\) chracterized by Fourier transforms representable as \(\widehat{\mu}(\pi) = e^{-\eta\left(\chi_\pi^{1/2}\right)}\cdot I_\pi\), where \(\eta \) denotes a second characteristic function of an infinitely divisible law on \(\mathbb{R}\), \(\pi\) denoting an irreducible representation. Various examples arise by subordination of symmetric central Gaussian laws. For the background on continuous convolution semigroups the reader is referred, e.g., to the monographs of \textit{H. Heyer} [Probability measures on locally compact groups. Berlin-Heidelberg-New York: Springer-Verlag (1977; Zbl 0376.60002)], and of the author [Lévy processes and stochastic calculus. 2nd ed. Cambridge: Cambridge University Press (2009; Zbl 1200.60001)]. Section 3 is concerned with regularity (\(L^2\) integrability, continuity or smoothness) of densities expressible as series \(k(\sigma) =\sum_\pi d_\pi\overline{c}_\pi\chi_\pi(\sigma)\) for almost all \(\sigma\in G\), where the Fourier transform is given as \(\widehat{\mu}(\pi)=c_\pi I_\pi \), and \(\chi_\pi\) denotes the character of \(\pi\). In analogy to the abelian case, regularity properties of the density \(k\) are described by regularity of the series \(\sum d_\pi…
reviewed by: Wilfried Hazod
Mathematics Subject Classification ID: 60B15
Mathematics Subject Classification ID: 60G51
Mathematics Subject Classification ID: 47D07
Mathematics Subject Classification ID: 43A05
zbMATH DE Number: 5995816
zbMATH Keywords: continuous convolution semigroups
zbMATH Keywords: central measures
zbMATH Keywords: compact Lie groups
zbMATH Keywords: pseudo differential operator
zbMATH Keywords: transition density
Wikidata QID: Q115240882
MaRDI profile type: Publication
arXiv ID: 1006.4711
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description: scientific article