Abstract
We describe certain sufficient conditions for an infinitely divisible probability measure on a Lie group to be embeddable in a continuous one-parameter semigroup of probability measures. A major class of Lie groups involved in the analysis consists of central extensions of almost algebraic groups by compactly generated abelian groups without vector part. This enables us in particular to conclude the embeddability of all infinitely divisible probability measures on certain connected Lie groups, including the so called Walnut group. The embeddability is concluded also under certain other conditions. Our methods are based on a detailed study of actions of certain nilpotent groups on special spaces of probability measures and on Fourier analysis along the fibering of the extension.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Chevalley C.: Theorie des Groupes de Lie. Hermann, Paris (1968)
Dani S.G.: On ergodic quasi-invariant measures of group automorphisms. Isr. J. Math. 43, 62–74 (1982)
Dani S.G.: On automorphism groups of connected Lie groups. Manuscr. Math. 74, 445–452 (1992)
Dani, S.G.: Asymptotic behaviour of measures under automorphisms. In: Probability Measures on Groups: Recent Directions and Trends, pp. 149–178. Tata Inst. Fund. Res., Mumbai (2006)
Dani S.G.: Convolution roots and embeddings of probability measures on locally compact groups. Indian J. Pure Appl. Math. 41, 241–250 (2010)
Dani S.G., McCrudden M.: On the factor sets of measures and local tightness of convolution semigroups over Lie groups. J. Theoret. Probab. 1, 357–370 (1988)
Dani S.G., McCrudden M.: Factors, roots and embeddability of measures on Lie groups. Math. Z. 199, 369–385 (1988)
Dani S.G., McCrudden M.: Embeddability of infinitely divisible distributions on linear Lie groups. Invent. Math. 110, 237–261 (1992)
Dani S.G., McCrudden M.: Convolution roots and embeddings of probability measures on Lie groups. Adv. Math. 209, 198–211 (2007)
Dani, S.G., McCrudden, M., Walker, S.: On the embedding problem for infinitely divisible distributions on certain Lie groups with toral center. Math. Z. 245, 781–790 (2003); Erratum: Math. Z. 252, 457–458 (2006)
Dani S.G., Schmidt Klaus: Affinely infinitely divisible distributions and the embedding problem. Math. Res. Lett. 9, 607–620 (2002)
Eichler, M., Zagier, D.: The theory of Jacobi forms. In: Progress in Mathematics, vol. 55. Birkhäuser, Boston (1985)
Heyer H.: Probability Measures on Locally Compact Groups. Springer, Berlin-New York (1977)
McCrudden M.: Infinitely divisible probabilities on SL(2, \({\mathbb {C}}\)) are continuously embedded. Math. Proc. Cambridge Philos. Soc. 92, 101–107 (1982)
McCrudden, M.: The embedding problem for probabilities on locally compact groups. In: Probability Measures on Groups: Recent Directions and Trends, pp. 331–363. Tata Inst. Fund. Res., Mumbai (2006)
Nahlus N.: Note on faithful representations and a local property of Lie groups. Proc. Am. Math. Soc. 125, 2767–2769 (1997)
Parthasarathy, K.R.: Probability Measures on Metric Spaces. AMS Chelsea Publishing, Providence (2005). Reprint of the 1967 original
Shah R.: The central limit problem on locally compact groups. Isr. J. Math. 110, 189–218 (1999)
Varadarajan, V.S.: Lie Groups, Lie Algebras, and Their Representations. Springer, New York (1984). Reprint of the 1974 edition
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to Mick McCrudden on the occasion of his 65th birthday.
Rights and permissions
About this article
Cite this article
Dani, S.G., Guivarc’h, Y. & Shah, R. On the embeddability of certain infinitely divisible probability measures on Lie groups. Math. Z. 272, 361–379 (2012). https://doi.org/10.1007/s00209-011-0937-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-011-0937-0