Inhomogeneous Lévy Processes in Homogeneous Spaces

Abstract

The martingale representation of inhomogeneous Lévy processes in a Lie group G in terms of an extended Lévy triple (b, A, η), obtained in Chapter 6, is extended to a homogeneous space X = G∕K in §8.1. The results are similar in form, but require a careful interpretation of certain operations on G∕K so that the formulae obtained on G, and their proofs, may be carried over to G∕K. We will also show that an inhomogeneous Lévy process in G∕K may be obtained as a projection of an inhomogeneous process in G. In §8.2, two special cases are considered. The first case is when the extended drift b _t on G∕K has a finite variation, then a more direct martingale representation may be obtained. The second case is an irreducible G∕K, such as a sphere, the representation takes an especially simple form in this case. The main results on G∕K may be proved in large part by essentially repeating the proofs on G, with a proper interpretation of group operations on G∕K. More details will be provided in §8.4. Some additional properties are considered in §8.3.