Lévy Processes in Compact Lie Groups

In this chapter, we apply Fourier analysis to study the distributions of Lévy processes in compact Lie groups. A similar study will be done for Lévy processes in symmetric spaces in the next chapter. After a brief review of the Fourier analysis on compact Lie groups, we discuss in §4.2 the Fourier expansion of the distribution density p t of a Lévy process g t in terms of matrix elements of irreducible unitary representations of G. It is shown that if g t has an L 2 density p t, then the Fourier series converges absolutely and uniformly on G, and the convergence to the uniform distribution (the normalized Haar measure) is obtained. In Section 4.3, for Lévy processes invariant under the inverse map, the L 2 density is shown to exist under a nondegenerate diffusion part or under an asymptotic condition on the Lévy measure, and the exponential convergence to the uniform distribution is obtained. The same results are proved in §4.4 for bi-invariant Lévy processes. In this case, the Fourier expansion is given in terms of irreducible characters, a more manageable form of Fourier series. Some examples are computed explicitly in the last section.