A Generalised Gangolli-Levy-Khintchine Formula for Infinitely Divisible Measures and Levy Processes on Semi-Simple Lie Groups and Symmetric Spaces

TL;DR

This paper extends the Lévy-Khintchine formula to characterize all infinitely divisible measures and Lévy processes on non-compact symmetric spaces using generalized spherical functions and representation theory.

Contribution

It generalizes Gangolli's formula to arbitrary infinitely divisible measures on symmetric spaces via Eisenstein integrals and representation theory.

Findings

Characterization of infinitely divisible measures on symmetric spaces.

Application to hyperbolic space example.

Extension of Fourier analysis tools to non-compact symmetric spaces.

Abstract

In 1964 R.Gangolli published a L\'{e}vy-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.