Compound Poisson Processes and Lévy Processes in Groups and Symmetric Spaces

A sample path description of compound Poisson processes on groups is given and applied to represent Lévy processes on connected Lie groups as almost sure limits of sequences of Brownian motions with drift interlaced with random jumps. We obtain spherically symmetric Lévy processes in Riemannian symmetric spaces of the form M=G\K, where G is a semisimple Lie group and K is a compact subgroup by projection of symmetric horizontal Lévy processes in G and give a straightforward proof of Gangolli's Lévy–Khintchine formula for their spherical transform. Finally, we show that such processes can be realized in Fock space in terms of creation, conservation, and annihilation processes. They appear as generators of a new class of factorizable representations of the current group C(R +, G). In the case where M is of noncompact type, these Fock space processes are indexed by the roots of G and the natural action of the Weyl group of G induces a (send quantized) unitary equivalence between them.