In this chapter, we consider the processes in a homogeneous space of a Lie group G induced by Lévy processes in G. In Section 2.1, these processes are introduced as one-point motions of Lévy processes in Lie groups. We derive the stochastic integral equations satisfied by these processes and discuss their Markov property. In Section 2.2, we consider the Markov processes in a homogeneous space of G that are invariant under the action of G. We study the relations among various invariance properties, we present Hunt's result on the generators of G-invariant processes, and we show that these processes are one-point motions of left Lévy processes in G that are also invariant under the right action of the isotropy subgroup. The last section of this chapter contains a discussion of Riemannian Brownian motions in Lie groups and homogeneous spaces.

One-Point Motions

Let G be a Lie group that acts on a manifold M on the left and let gt be a process in G. For any x ∈ M, the process xt = gtx will be called the one-point motion of gt in M starting from x. In general, the one-point motion of a Markov process in G is not a Markov process in M.