Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

A generalised Gangolli–Lévy–Khintchine formula for infinitely divisible measures and Lévy processes on semi-simple Lie groups and symmetric spaces

David Applebaum and Anthony Dooley

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In 1964 R. Gangolli published a Lévy–Khintchine type formula which characterised $K$-bi-invariant infinitely divisible probability measures on a symmetric space $G/K$. His main tool was Harish-Chandra’s spherical functions which he used to construct a generalisation of the Fourier transform of a measure. In this paper we use generalised spherical functions (or Eisenstein integrals) and extensions of these which we construct using representation theory to obtain such a characterisation for arbitrary infinitely divisible probability measures on a non-compact symmetric space. We consider the example of hyperbolic space in some detail.


R. Gangolli (1964) publia une formule du type Lévy–Khintchine, caractérisant les probabilités infiniment divisibles $K$-bi-invarantes sur un espace symétrique $G/K$. Son outil principal fut les fonctions sphériques de Harish-Chandra qu’il utilisa pour construire une généralisation de la transformée de Fourier d’une mesure. Dans cet article, on se sert des fonctions sphériques généralisées (les intégrales d’Eisenstein) de leurs généralisations, que l’on construit à partir de la théorie de représentations, pour obtenir une telle caractérisation pour les probabilités quelquonques infiniment divisibles sur un espace symétrique non-compact. On considère, en détail, le cas de l’espace hyperbolique.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 51, Number 2 (2015), 599-619.

First available in Project Euclid: 10 April 2015

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Primary: 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization 60E07: Infinitely divisible distributions; stable distributions 43A30: Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc. 60G51: Processes with independent increments; Lévy processes 22E30: Analysis on real and complex Lie groups [See also 33C80, 43-XX] 53C35: Symmetric spaces [See also 32M15, 57T15] 43A05: Measures on groups and semigroups, etc.

Lévy process Lie group Lie algebra Generalised Eisenstein integral Eisenstein transform Extended Gangolli Lévy–Khintchine formula Symmetric space Hyperbolic space


Applebaum, David; Dooley, Anthony. A generalised Gangolli–Lévy–Khintchine formula for infinitely divisible measures and Lévy processes on semi-simple Lie groups and symmetric spaces. Ann. Inst. H. Poincaré Probab. Statist. 51 (2015), no. 2, 599--619. doi:10.1214/13-AIHP570.

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