On the embeddability of certain infinitely divisible probability measures on Lie groups
type: Item
instance of: scholarly article
title: On the embeddability of certain infinitely divisible probability measures on Lie groups
zbMATH Open document ID: 1261.60007
author: Yves Guivarc'h
author: Riddhi Shah
author: S. G. Dani
published in: Mathematische Zeitschrift
publication date: 2012-10-22T00:00:00Z
full work available at URL: https://arxiv.org/abs/1102.1639
review text: The embedding problem for infinitely divisible probabilities on locally compact groups was one of the most challenging problems during the last decades. The investigations started with pioneer papers, e.g., by \textit{E. Siebert} [Z. Wahrscheinlichkeitstheor. Verw. Geb. 28, 227--247 (1974; Zbl 0264.60004)], followed by a series of investigations by S. G. Dani and M. McCrudden (see, e.g., [\textit{M. McCrudden}, in: S. G. Dani (ed.) et al., Probability measures on groups: recent directions and trends. Tata Institute of Fundamental Research 18, 331--363 (2006; Zbl 1145.60006); \textit{S. G. Dani}, Indian J. Pure Appl. Math. 41, No. 1, 241--250 (2010; Zbl 1200.60010)] and the references therein). It is a natural conjecture that the embedding problem has a positive solution for all connected Lie groups, but, for a long time, it was only possible to show this for connected Lie groups with nil-radical containing no torus of positive dimension. An example of a group not belonging to that class is the ``walnut group'' \(W\), a four dimensional Lie group, (cf., e.g., [\textit{S. G. Dani}, \textit{M. McCrudden} and \textit{S. Walker}, Math. Z. 245, No. 4, 781--790 (2003); erratum ibid. 252, No. 2, 457--458 (2006; Zbl 1050.22009)] for details). In the paper under review, the authors solve the embedding problem for a class of groups containing \(W\) (\(W\) was called walnut group because it seemed to be very hard to solve the embedding problem for that group) under an additional condition on the measure. Precisely, embeddability is shown for Lie groups \(G\) admitting a surjective continuous homomorphism \(p\) onto an almost algebraic group with \(\operatorname{ker} p \subseteq \operatorname{cent}(G)\), \(T:= \left(\operatorname{ker} p\right)^0\) being compact, and for infinitely divisible probabilities \(\mu\) whose image on \(G/T\) has no non-trivial idempotent factor (Theorem 1.3). The proof relies on standard techniques and methods used in the afore-mentioned investigations…
reviewed by: Wilfried Hazod
Mathematics Subject Classification ID: 60B15
Mathematics Subject Classification ID: 43A05
Mathematics Subject Classification ID: 22E30
Mathematics Subject Classification ID: 60E07
zbMATH DE Number: 6097111
zbMATH Keywords: embedding problem
zbMATH Keywords: infinitely divisible laws
zbMATH Keywords: walnut group
zbMATH Keywords: rational embedding
MaRDI profile type: Publication
OpenAlex ID: W1997979755
Wikidata QID: Q115388782
arXiv ID: 1102.1639
cites work: Q5580320
cites work: On ergodic quasi-invariant measures of group automorphism
cites work: On automorphism groups of connected Lie groups
cites work: Q3509394
cites work: Convolution roots and embeddings of probability measures on locally compact groups
cites work: On the factor sets of measures and local tightness of convolution semigroups over Lie groups
cites work: Factors, roots and embeddability of measures on Lie groups
cites work: Embeddability of infinitely divisible distributions on linear Lie groups
cites work: Convolution roots and embeddings of probability measures on Lie groups
cites work: On the embedding problem for infinitely divisible distributions on certain Lie groups with toral center
cites work: Affinely infinitely divisible distributions and the embedding problem
cites work: The theory of Jacobi forms
cites work: Q4153376
cites work: Infinitely divisible probabilities on SL(2,) are continuously embedded
cites work: Q3509397
cites work: Note on faithful representations and a local property of Lie groups
cites work: Q5533878
cites work: The central limit problem on locally compact groups
cites work: Q4527707
DOI: 10.1007/S00209-011-0937-0
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description: scientific article; zbMATH DE number 6097111