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

Applebaum, David; Dooley, Anthony

Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 2, pp. 599-619.

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.

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.

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