Lévy processes and Fourier analysis on compact Lie groups.(English)Zbl 1054.60003

Let $$G$$ be a compact Lie group. G-valued Lévy process $$(g(t)$$, $$t \geq 0)$$ is a stochastic process such that for any $$s < u \leq v < t$$, $$g(s)^{-}g(u)$$ and $$g(v)^{-}g(t)$$ are independent and have the same distribution law. The purpose of the paper is to show the exponential convergence of the distrubution law of $$g(t)$$ to the normalized Haar measure. In Fourier analysis on compact Lie groups, the Weyl theorem is a fundamental tool. In 1956, G. A. Hunt gave the formula of semigroup generating functions of Lévy process whose terms are elliptic terms, constant terms and jumping terms with Lévy measure. Using this formula, with the Weyl theorem the analysis is done. A Lévy process is called conjugate invariant if its probability law $$\mu(t)$$ is invariant under the $$\text{Ad}(g),g \in G$$, action. Under some hypothesis about elliptic part, it is shown that $$\mu(t)$$ of a conjugate invariant Lévy process has a density in $$L^{2}(G)$$ and it is related to characters of irreducible representations.

MSC:

 60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
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References:

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