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Stable Lévy processes on nilpotent Lie groups. (English) Zbl 0814.60003

Kunita, Hiroshi (ed.) et al., Stochastic analysis on infinite dimensional spaces. Proceedings of the U.S.-Japan bilateral seminar, January 4-8, 1994, Baton Rouge, LA, USA. Harlow, Essex: Longman Scientific Technical. Pitman Res. Notes Math. Ser. 310, 167-182 (1994).
For a Lie group \(G\) admitting dilations \(J_ r\) (i.e., \(J_ r\), \(r>0\), is a one-parameter group of automorphisms of \(G)\) the author introduces the notion of \(J\)-stable process \(\varphi_ t\), \(t>0\), on \(G\) by the equation \(J_ r \varphi_ t = \varphi_{r \cdot t}\) for \(r,t>0\). In case of Euclidean space as \(G\) one gets (strictly) operator-stable processes. The main result is that every Lévy process on a nilpotent Lie group \(G\) is obtained by integrating a Lévy process on the associated Lie algebra \({\mathcal G}\). On the other hand, not all strictly operator-stable processes on a Lie algebra generate a stable process on the corresponding Lie group.
For the entire collection see [Zbl 0801.00037].
Reviewer: Z.Jurek (Wrocław)

MSC:

60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
60E07 Infinitely divisible distributions; stable distributions
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