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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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[1] Applebaum, D. (2000a). \levy processes in stochastic differential geometry. In \levy Processes : Theory and Applications (O. Barnsdorff-Nielsen, T. Mikosch and S. Resnick, eds.) 111–139. Birkhäuser, Boston.
[2] Applebaum, D. (2000b). Compound Poisson processes and \levy processes in groups and symmetric spaces. J. Theoret. Probab. 13 383–425. · Zbl 0985.60047
[3] Azencott, R., Baldi, P., Bellaiche, A., Bellaiche, C., Bougerol, P., Chaleyat-Maurel, M., Elie, L. and Granara, J. (1981). Géodésiques et diffusions en temps petit. Asterisque 84–85.
[4] Bröcker, T. and Dieck, T. (1985). Representations of Compact Lie Groups . Springer, New York. · Zbl 0581.22009
[5] Chatelin, F. (1993). Eigenvalues of Matrices . Wiley, New York. · Zbl 0783.65031
[6] Diaconis, P. (1988). Group Representations in Probability and Statistics . IMS, Hayward, CA. · Zbl 0695.60012
[7] Gangolli, R. (1964). Isotropic infinitely divisible measures on symmetric spaces. Acta Math. 111 213–246. · Zbl 0154.43804
[8] Helgason, S. (2000). Groups and geometric analysis. Amer. Math. Soc. Collog. Publ. · Zbl 0965.43007
[9] Heyer, H. (1968). L’analyse de Fourier non-commutative et applications à la théorie des probabilitités. Ann. Inst. H. Poincaré Sect. B ( N.S. ) 143–164. · Zbl 0165.19102
[10] Hunt, G. A. (1956). Semigroup of measures on Lie groups. Trans. Amer. Math. Soc. 81 264–293. · Zbl 0073.12402
[11] Klyachko, A. A. (2000). Random walks on symmetric spaces and inequalities for matrix spectra. Linear Algebra Appl. 319 37–59. · Zbl 0980.15015
[12] Kobayashi, S. and Nomizu, K. (1963). Foundations of Differential Geometry I . Interscience Publishers. · Zbl 0119.37502
[13] Liao, M. (1998). Lévy processes in semi-simple Lie groups and stability of stochastic flows. Trans. Amer. Math. Soc. 350 501–522. · Zbl 0909.60061
[14] Liao, M. (2002). Dynamical properties of \levy processes in Lie groups. Stoch. Dyn. 2 1–23. · Zbl 1002.58017
[15] Rosenthal, J. S. (1994). Random rotations: Characters and random walks on SO(N). Ann. Probab. 22 398–423. JSTOR: · Zbl 0799.60007
[16] Siebert, E. (1981). Fourier analysis and limit theorems for convolution semigroups on a locally compact groups. Adv. in Math. 39 111–154. · Zbl 0469.60014
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