This chapter contains an introduction to Lévy processes in a general Lie group. The left and right Lévy processes in a topological group G are defined in Section 1.1. They can be constructed from a convolution semigroup of probability measures on G and are Markov processes with left or right invariant Feller transition semigroups. In the next two sections, we introduce Hunt's theorem for the generator of a Lévy process in a Lie group G and prove some related results for the Lévy measure determined by the jumps of the process. In Section 1.4, the Lévy process is characterized as a solution of a stochastic integral equation driven by a Brownian motion and an independent Poisson random measure whose characteristic measure is the Lévy measure. Some variations and extensions of this stochastic integral equation are discussed. The proofs of the stochastic integral equation characterization, due to Applebaum and Kunita, and of Hunt's theorem, will be given in Chapter 3. For Lévy processes in matrix groups, a more explicit stochastic integral equation, written in matrix form, is obtained in Section 1.5.

Lévy Processes

The reader is referred to Appendices A and B for the basic definitions and facts on Lie groups, stochastic processes, and stochastic analysis.