Brownian motion and Lévy processes in locally compact groups - Applebaum, D. (Methods of Functional Analysis and Topology, 2006)
It is shown that every L\’evy process on a locally compact group G is determined by a sequence of one-dimensional Brownian motions and an independent Poisson random measure. As a consequence, we are able to give a very straightforward proof of sample path continuity for Brownian motion in G. We also show that every L\’evy process on G is of pure jump type, when G is totally disconnected.