Decomposition of Markov Processes

Abstract

We have studied the decomposition of an invariant Markov process, under a non-transitive action of a topological group, into a radial and an angular parts in §1.5 and §1.6. In this chapter, this decomposition will be studied in light of the representation theory of inhomogeneous Lévy processes in the homogeneous spaces of Lie groups developed in the previous chapter. In §9.1, the extended Lévy triple used in the representation of an inhomogeneous Lévy process will be extended to a functional form, and the decomposition of an invariant Markov processes will be described in terms of this functional triple. In §9.2, for a continuous invariant Markov process with irreducible orbits, we establish a skew-product decomposition into a radial motion and an independent angular motion with a time change, which extends the well-known skew-product of Brownian motion in $\mathbb {R}^n$. §9.3 and §9.4 are devoted to a more detailed study of invariant diffusion processes. As a simple application, in §9.5, we study a diffusion process in a Euclidean space that is invariant under the translations in a subspace. In §9.6, we consider a simple example with jumps.