×
Applebaum, David; Dooley, Anthony

A generalised Gangolli-Lévy-Khintchine formula for infinitely divisible measures and Lévy processes on semi-simple Lie groups and symmetric spaces. (English. French summary) Zbl 1353.60007

Ann. Inst. Henri Poincaré, Probab. Stat. 51, No. 2, 599-619 (2015).
Let \(G\) be a connected semisimple Lie group and \(K\) a compact subgroup of \(G\). In [Acta Math. 111, 213–246 (1964; Zbl 0154.43804)], R. Gangolli found an analogue of the well-known Lévy-Khinchin formula for \(K\)-bi-invariant infinitely divisible measures \(\mu\) on \(G\). Gangolli’s proof is based on the theory of Harish-Chandra’s spherical functions which is used to construct a generalization of the Fourier transform of a measure. The authors extend Gangolli’s result to generally infinitely divisible measures on non-compact symmetric space without bi-invariance assumption. This problem is more complicated. Their approach is different from Gangolli’s and use the construction of generalised Eisenstein integrals. Some applications are also considered.
Reviewer: Gennadiy M. Fel’dman (Khar’kov)

Cited in 1 Document

MSC:

60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
60E07 Infinitely divisible distributions; stable distributions
43A30 Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.

Keywords:

Lie group; generalised Eisenstein integral; Lévy-Khinchin formula; symmetric space

Citations:

Zbl 0154.43804
PDFBibTeX XMLCite
\textit{D. Applebaum} and \textit{A. Dooley}, Ann. Inst. Henri Poincaré, Probab. Stat. 51, No. 2, 599--619 (2015; Zbl 1353.60007)
Full Text: DOI arXiv Euclid

References:

[1] S. Albeverio and M. Gordina. Lévy processes and their subordination in matrix Lie groups. Bull. Sci. Math. 131 (2007) 738-760. · Zbl 1140.60006 · doi:10.1016/j.bulsci.2006.11.004
[2] D. Applebaum. Compound Poisson processes and Lévy processes in groups and symmetric spaces. J. Theoret. Probab. 13 (2000) 383-425. · Zbl 0985.60047 · doi:10.1023/A:1007845508326
[3] D. Applebaum. On the subordination of spherically symmetric Lévy processes in Lie groups. Internat. Math. J. 1 (2002) 185-195. · Zbl 0984.60018
[4] D. Applebaum. Lévy Processes and Stochastic Calculus , 2nd edition. Cambridge Univ. Press, Cambridge, 2009. · Zbl 1200.60001 · doi:10.1017/CBO9780511809781
[5] D. Applebaum. Aspects of recurrence and transience for Lévy processes in transformation groups and non-compact Riemannian symmetric pairs. J. Australian Math. Soc. 94 (2013) 304-320. · Zbl 1277.60011 · doi:10.1017/S1446788713000062
[6] D. Applebaum and A. Estrade. Isotropic Lévy processes on Riemannian manifolds. Ann. Probab. 28 (2000) 166-184. · Zbl 1044.60035 · doi:10.1214/aop/1019160116
[7] J. Arthur. A Paley-Wiener theorem for real reductive groups. Acta Math. 150 (1983) 1-89. · Zbl 0514.22006 · doi:10.1007/BF02392967
[8] C. Berg. Dirichlet forms on symmetric spaces. Ann. Inst. Fourier (Grenoble) 23 (1973) 135-156. · Zbl 0243.31013 · doi:10.5802/aif.448
[9] C. Berg and J. Faraut. Semi-groupes de Feller invariants sur les espaces homogènes non moyennables. Math. Z. 136 (1974) 279-290. · Zbl 0267.43004 · doi:10.1007/BF01213872
[10] W. R. Bloom and H. Heyer. Harmonic Analysis of Probability Measures on Hypergroups . de Gruyter, Berlin, 1995. · Zbl 0828.43005
[11] S. G. Dani and M. McCrudden. Embeddability of infinitely divisible distributions on linear Lie groups. Invent. Math. 110 (1992) 237-261. · Zbl 0771.60007 · doi:10.1007/BF01231332
[12] S. G. Dani and M. McCrudden. Convolution roots and embedding of probability measures on Lie groups. Adv. Math. 209 (2007) 198-211. · Zbl 1124.60009 · doi:10.1016/j.aim.2006.05.002
[13] R. Gangolli. Isotropic infinitely divisible measures on symmetric spaces. Acta Math. 111 (1964) 213-246. · Zbl 0154.43804 · doi:10.1007/BF02391013
[14] R. Gangolli. Sample functions of certain differential processes on symmetric spaces. Pacific J. Math. 15 (1965) 477-496. · Zbl 0141.14903 · doi:10.2140/pjm.1965.15.477
[15] R. K. Getoor. Infinitely divisible probabilities on the hyperbolic plane. Pacific J. Math. 11 (1961) 1287-1308. · Zbl 0124.34502 · doi:10.2140/pjm.1961.11.1287
[16] S. Helgason. Groups and Geometric Analysis . Academic Press, New York, 1984. Reprinted with corrections by the Amer. Math. Soc., Providence, RI, 2000.
[17] S. Helgason. Geometric Analysis on Symmetric Spaces . Amer. Math. Soc., Providence, RI, 1994. · Zbl 0809.53057
[18] H. Heyer. Convolution semigroups of probability measures on Gelfand pairs. Expo. Math. 1 (1983) 3-45. · Zbl 0517.60004
[19] H. Heyer. Transient Feller semigroups on certain Gelfand pairs. Bull. Inst. Math. Acad. Sinica 11 (1983) 227-256. · Zbl 0523.60013
[20] S. F. Huckemann, P. T. Kim, J.-Y. Koo and A. Munk. Möbius deconvolution on the hyperbolic plane with application to impedance density estimation. Ann. Statist. 38 (2010) 2465-2498. · Zbl 1203.62055 · doi:10.1214/09-AOS783
[21] G. A. Hunt. Semigroups of measures on Lie groups. Trans. Amer. Math. Soc. 81 (1956) 264-293. · Zbl 0073.12402 · doi:10.2307/1992917
[22] A. W. Knapp. Representation Theory of Semisimple Groups . Princeton Univ. Press, Princeton, NJ, 1986. · Zbl 0604.22001
[23] A. W. Knapp. Lie Groups Beyond an Introduction , 2nd edition. Birkhäuser, Berlin, 2002. · Zbl 1075.22501
[24] M. Liao. Lévy Processes in Lie Groups . Cambridge Univ. Press, Cambridge, 2004.
[25] M. Liao and L. Wang. Lévy-Khinchin formula and existence of densities for convolution semigroups on symmetric spaces. Potential Anal. 27 (2007) 133-150. · Zbl 1127.58032 · doi:10.1007/s11118-007-9048-2
[26] G. Ólafsson and H. Schlichtkrull. Representation theory, Radon transform and the heat equation on a Riemannian symmetric space. In Group Representations, Ergodic Theory, and Mathematical Physics: A Tribute to George W. Mackey 315-344. Contemp. Math. 449 . Amer. Math. Soc., Providence, RI, 2008. · Zbl 1167.33309 · doi:10.1090/conm/449/08718
[27] K.-I. Sato. Lévy Processes and Infinite Divisibility . Cambridge Univ. Press, Cambridge, 1999.
[28] R. L. Schilling. Conservativeness and extensions of Feller semigroups. Positivity 2 (1998) 239-256. · Zbl 0919.47033 · doi:10.1023/A:1009748105208
[29] E. P. van den Ban and H. Schlichtkrull. The Plancherel decomposition for a reductive symmetric space. II. Representation theory. Invent. Math. 161 (2005) 567-628. · Zbl 1078.22013 · doi:10.1007/s00222-004-0432-x
[30] E. P. van den Ban. The principal series for a reductive symmetric space. II. Eisenstein integrals. J. Funct. Anal. 109 (1992) 331-441. · Zbl 0791.22008 · doi:10.1016/0022-1236(92)90021-A
[31] E. P. van den Ban. Weyl eigenfunction expansions and harmonic analysis on non-compact symmetric spaces. In Groups and Analysis 24-62. London Math. Soc. Lecture Note Ser. 354 . Cambridge Univ. Press, Cambridge, 2008. · Zbl 1176.22013 · doi:10.1017/CBO9780511721410.002
[32] E. P. van den Ban. Private e-mail communication to the authors.
[33] N. R. Wallach. Real Reductive Groups. I . Academic Press, Boston, MA, 1988. · Zbl 0666.22002
[34] H. Zhang. Lévy stochastic differential geometry with applications in derivative pricing. Ph.D. thesis, Univ. New South Wales, 2010.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.