 Lévy Processes in Compact Lie GroupsMing Liao
Chapter Chapter 4 in Invariant Markov Processes Under Lie Group Actions, 2018, pp 103-133 from Springer Abstract: Abstract 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. Date: 2018 There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it. Related works: Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-319-92324-6_4 Ordering information: This item can be ordered from DOI: 10.1007/978-3-319-92324-6_4 Access Statistics for this chapter More chapters in Springer Books from Springer |
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