Invariant Markov Processes Under Actions of Topological Groups

Although the main purpose of this work is to study invariant Markov processes under actions of Lie groups, in this first chapter we will develop the basic definitions and properties under the more general action of a topological group which is assumed to be locally compact. A special case is a Markov process in a topological group G invariant under the action of G on itself by left translations. Such a process may be characterized by independent and stationary increments, and will be called a Lévy process in G, see Section §1.2. A more general case is a Markov process in a topological space X under the transitive action of a topological group G. Such a process may be regarded as a process in a topological homogeneous space G∕K of G. We will develop some basic properties under the assumption that K is compact. Because such a process may be characterized by independent and stationary increments in a certain sense, so will be called a Lévy process in G∕K, see §1.3. These results may be extended to time inhomogeneous invariant Markov processes, called inhomogeneous Lévy processes, see §1.4. In all these cases, the distribution of the process is characterized by a convolution semigroup of probability measures on G or on G∕K. It is shown in §1.5 and §1.6 that an invariant Markov process under a non-transitive action, under some suitable assumptions, may be decomposed into a radial part and an angular part. The radial part can be an arbitrary Markov process in a subspace that is transversal to the orbits of the group action, and under the conditional distribution given the radial process, the angular part is an inhomogeneous Lévy process in the standard orbit.