Invariant Markov processes under actions of Lie groups

Abstract

We present a representation theoryfor invariant Markov processes under Lie group actions, in the spirit of the classic levy-Khinchin representation. We will study invariant Markov processes at three different levels of generality.First, we consider Markov processes in Lie groups that are invariant under translations, and then Markov processes in manifolds that are invariant under transitive group actions.These two types of processes are called levy processes, and they possess a triple representation. The third type of processes are Markov processes that are invariant under non-transitive group actions. Under certain conditions, such a process may be decomposed into a radial part, that is transversal to group orbits, and an angular part, that is along an orbit.The latter is a time inhomogeneous invariant Markov process, and may be represented by a time-dependent triple.