||A Levy process in a Lie group is a process that possesses independent and stationary multiplicative increments. The theory of such processes is not merely an extension of Levy processes in Euclidean spaces. Because of the unique structures possessed by non-commutative Lie groups, these processes exhibit certain interesting properties which are not present for their counterparts in Euclidean spaces.
The concept of Levy processes may be extended to include Markov processes in a homogeneous space that are invariant under the group action. More generally, we will also study processes in Lie groups and homogeneous spaces that possess independent, but not necessarily stationary, increments, called nonhomogeneous Levy processes. These processes appear naturally when studying a decomposition of a general Markov process in a manifold invariant under a group action. This extends the skew-product decomposition of Brownian motion, but in the present general form, it appears new even on Euclidean spaces. In these notes, we will provide an introduction to Levy processes in Lie groups and homogeneous spaces, and present some selected results in this area. The reader is referred to the literature for the most of proofs, but some explanation will be given to the results not yet published.