A generalised Gangolli–Lévy–Khintchine formula for infinitely divisible measures and Lévy processes on semi-simple Lie groups and symmetric spaces
Annales de l'I.H.P. Probabilités et statistiques, Volume 51 (2015) no. 2, p. 599-619

In 1964 R. Gangolli published a Lévy–Khintchine type formula which characterised K-bi-invariant infinitely divisible probability measures on a symmetric space G/K. His main tool was Harish-Chandra’s spherical functions which he used to construct a generalisation of the Fourier transform of a measure. In this paper we use generalised spherical functions (or Eisenstein integrals) and extensions of these which we construct using representation theory to obtain such a characterisation for arbitrary infinitely divisible probability measures on a non-compact symmetric space. We consider the example of hyperbolic space in some detail.

R. Gangolli (1964) publia une formule du type Lévy–Khintchine, caractérisant les probabilités infiniment divisibles K-bi-invarantes sur un espace symétrique G/K. Son outil principal fut les fonctions sphériques de Harish-Chandra qu’il utilisa pour construire une généralisation de la transformée de Fourier d’une mesure. Dans cet article, on se sert des fonctions sphériques généralisées (les intégrales d’Eisenstein) de leurs généralisations, que l’on construit à partir de la théorie de représentations, pour obtenir une telle caractérisation pour les probabilités quelquonques infiniment divisibles sur un espace symétrique non-compact. On considère, en détail, le cas de l’espace hyperbolique.

DOI : https://doi.org/10.1214/13-AIHP570
Classification:  60B15,  60E07,  43A30,  60G51,  22E30,  53C35,  43A05
Keywords: lévy process, Lie group, Lie algebra, generalised Eisenstein integral, Eisenstein transform, extended Gangolli Lévy–Khintchine formula, symmetric space, hyperbolic space
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     author = {Applebaum, Dave and Dooley, Anthony},
     title = {A generalised Gangolli--L\'evy--Khintchine formula for infinitely divisible measures and L\'evy processes on semi-simple Lie groups and symmetric spaces},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {51},
     number = {2},
     year = {2015},
     pages = {599-619},
     doi = {10.1214/13-AIHP570},
     mrnumber = {3335018},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2015__51_2_599_0}
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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. Annales de l'I.H.P. Probabilités et statistiques, Volume 51 (2015) no. 2, pp. 599-619. doi : 10.1214/13-AIHP570. http://www.numdam.org/item/AIHPB_2015__51_2_599_0/

[1] S. Albeverio and M. Gordina. Lévy processes and their subordination in matrix Lie groups. Bull. Sci. Math. 131 (2007) 738–760. | MR 2372464 | Zbl 1140.60006

[2] D. Applebaum. Compound Poisson processes and Lévy processes in groups and symmetric spaces. J. Theoret. Probab. 13 (2000) 383–425. | MR 1777540 | Zbl 0985.60047

[3] D. Applebaum. On the subordination of spherically symmetric Lévy processes in Lie groups. Internat. Math. J. 1 (2002) 185–195. | MR 1828652 | Zbl 0984.60018

[4] D. Applebaum. Lévy Processes and Stochastic Calculus, 2nd edition. Cambridge Univ. Press, Cambridge, 2009. | MR 2512800 | Zbl 1073.60002

[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. | MR 3110816 | Zbl 1277.60011

[6] D. Applebaum and A. Estrade. Isotropic Lévy processes on Riemannian manifolds. Ann. Probab. 28 (2000) 166–184. | MR 1756002 | Zbl 1044.60035

[7] J. Arthur. A Paley–Wiener theorem for real reductive groups. Acta Math. 150 (1983) 1–89. | MR 697608 | Zbl 0514.22006

[8] C. Berg. Dirichlet forms on symmetric spaces. Ann. Inst. Fourier (Grenoble) 23 (1973) 135–156. | Numdam | MR 393531 | Zbl 0243.31013

[9] C. Berg and J. Faraut. Semi-groupes de Feller invariants sur les espaces homogènes non moyennables. Math. Z. 136 (1974) 279–290. | MR 353406 | Zbl 0267.43004

[10] W. R. Bloom and H. Heyer. Harmonic Analysis of Probability Measures on Hypergroups. de Gruyter, Berlin, 1995. | MR 1312826 | 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. | MR 1185583 | Zbl 0771.60007

[12] S. G. Dani and M. Mccrudden. Convolution roots and embedding of probability measures on Lie groups. Adv. Math. 209 (2007) 198–211. | MR 2294221 | Zbl 1124.60009

[13] R. Gangolli. Isotropic infinitely divisible measures on symmetric spaces. Acta Math. 111 (1964) 213–246. | MR 161350 | Zbl 0154.43804

[14] R. Gangolli. Sample functions of certain differential processes on symmetric spaces. Pacific J. Math. 15 (1965) 477–496. | MR 185654 | Zbl 0141.14903

[15] R. K. Getoor. Infinitely divisible probabilities on the hyperbolic plane. Pacific J. Math. 11 (1961) 1287–1308. | MR 133858 | Zbl 0124.34502

[16] S. Helgason. Groups and Geometric Analysis. Academic Press, New York, 1984. Reprinted with corrections by the Amer. Math. Soc., Providence, RI, 2000. | MR 1790156 | Zbl 0965.43007

[17] S. Helgason. Geometric Analysis on Symmetric Spaces. Amer. Math. Soc., Providence, RI, 1994. | MR 1280714 | Zbl 1157.43003

[18] H. Heyer. Convolution semigroups of probability measures on Gelfand pairs. Expo. Math. 1 (1983) 3–45. | MR 693806 | Zbl 0517.60004

[19] H. Heyer. Transient Feller semigroups on certain Gelfand pairs. Bull. Inst. Math. Acad. Sinica 11 (1983) 227–256. | MR 723027 | 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. | MR 2676895 | Zbl 1203.62055

[21] G. A. Hunt. Semigroups of measures on Lie groups. Trans. Amer. Math. Soc. 81 (1956) 264–293. | MR 79232 | Zbl 0073.12402

[22] A. W. Knapp. Representation Theory of Semisimple Groups. Princeton Univ. Press, Princeton, NJ, 1986. | MR 855239 | Zbl 0993.22001

[23] A. W. Knapp. Lie Groups Beyond an Introduction, 2nd edition. Birkhäuser, Berlin, 2002. | MR 1920389 | Zbl 1075.22501

[24] M. Liao. Lévy Processes in Lie Groups. Cambridge Univ. Press, Cambridge, 2004. | MR 2060091 | Zbl 1076.60004

[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. | MR 2322502 | Zbl 1127.58032

[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. | MR 2391810 | Zbl 1167.33309

[27] K.-I. Sato. Lévy Processes and Infinite Divisibility. Cambridge Univ. Press, Cambridge, 1999. | Zbl 0973.60001

[28] R. L. Schilling. Conservativeness and extensions of Feller semigroups. Positivity 2 (1998) 239–256. | MR 1653474 | Zbl 0919.47033

[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. | MR 2181716 | Zbl 1078.22013

[30] E. P. Van Den Ban. The principal series for a reductive symmetric space. II. Eisenstein integrals. J. Funct. Anal. 109 (1992) 331–441. | MR 1186325 | Zbl 0791.22008

[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. | MR 2528460 | Zbl 1176.22013

[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. | MR 929683 | Zbl 0785.22001

[34] H. Zhang. Lévy stochastic differential geometry with applications in derivative pricing. Ph.D. thesis, Univ. New South Wales, 2010.