Lévy Processes in Lie Groups

Abstract

It is well known that the distribution of a classical Lévy process in a Euclidean space $\mathbb {R}^d$ is determined by a triple of a drift vector, a covariance matrix, and a Lévy measure, which are called the characteristics of the Lévy process. The triple appears in the Lévy-Khinchin formula, which is the Fourier transform of the distribution, or in the pathwise Lévy-Itô representation. In the latter representation, the three elements of the triple correspond respectively to a nonrandom drift, a diffusion part, and a pure jump part of the process. A Lévy process in a Lie group G cannot be decomposed into three parts as in $\mathbb {R}^d$, due to the non-commutative nature of G, but the triple representation holds in an infinitesimal sense, in the form of Hunt’s generator formula, to be discussed in §2.1. The relation between Lévy measures and jumps of Lévy processes is considered in §2.2.