Like the simple additive structure on an Euclidean space, the more complicated algebraic structure on a Lie group provides a convenient setting under which various stochastic processes with interesting properties may be defined and studied. An important class of such processes are Lévy processes that possess translation invariant distributions. Since a Lie group is in general noncommutative, there are two different types of Lévy processes, left and right Lévy processes, defined respectively by the left and right translations. Because the two are in natural duality, for most purposes, it suffices to study only one of them and derive the results for the other process by a simple transformation. However, the two processes play different roles in applications. Note that a Lévy process may also be characterized as a process that possesses independent and stationary increments.

The theory of Lévy processes in Lie groups is not merely an extension of the theory of Lévy processes in Euclidean spaces. Because of the unique structures possessed by the noncommutative Lie groups, these processes exhibit certain interesting properties that are not present for their counterparts in Euclidean spaces. These properties reveal a deep connection between the behavior of the stochastic processes and the underlying algebraic and geometric structures of the Lie groups.