4 - Lévy Processes in Compact Lie Groups and Fourier Analysis

Summary

In this chapter, we apply Fourier analysis to study the distributions of Lévy processes g_t in compact Lie groups. After a brief review of the Fourier analysis on compact Lie groups based on the Peter—Weyl theorem, we discuss in Section 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 L² density p_t, then the Fourier series converges absolutely and uniformly on G, and the coefficients tend to 0 exponentially as time t → ∞. In Section 4.3, for Lévy processes invariant under the inverse map, the L² distribution density is shown to exist, and the exponential bounds for the density as well as the exponential convergence of the distribution to the normalized Haar measure are obtained. The same results are proved in Section 4.4 for conjugate invariant Lévy processes. In this case, the Fourier expansion is given in terms of irreducible characters, a more manageable form of Fourier series. An example on the special unitary group SU(2) is computed explicitly in the last section. The results of this chapter are taken from Liao [43].

Fourier Analysis on Compact Lie Groups

This section is devoted to a brief discussion of Fourier series of L² functions on a compact Lie group G based on the Peter—Weyl theorem.