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Unicité du plongement d’une mesure de probabilité dans un semi-groupe de convolution gaussien. Cas non-abélien. (French) Zbl 0562.60010

The author proves the following remarkable theorem: Let G be a simply connected step two nilpotent Lie group and let \(\mu\) be a probability measure on G. Then there exists at most one Gaussian convolution semigroup \((\nu_ t)_{t>0}\) on G such that \(\nu_ 1=\mu.\)
If \(G={\mathbb{R}}^ n\) this result is classical. In the present (non- Abelian) situation the proof relies on the following fact: the diffusion process \(\gamma\) associated with a Gaussian semigroup on G can be constructed by a certain recurrence procedure, cf. B. Roynette, z. Wahrscheinlichkeitstheor. Verw. Geb. 32, 133-138 (1975; Zbl 0312.60036).
Reviewer’s remark: It would be interesting to have an answer to the following related question: Given a Gaussian measure \(\mu\) on G can there exists a non-Gaussian convolution semigroup \((\mu_ t)_{t>0}\) on G such that \(\mu_ 1=\mu\)?
Reviewer: E.Siebert

MSC:

60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
43A05 Measures on groups and semigroups, etc.

Citations:

Zbl 0312.60036
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References:

[1] Drisch, Th., Gallardo, L.: Stable laws on the Heisenberg groups. Dans: Probability measures on groups VII. Lecture Notes in Math., Vol. 1064, pp. 56-79. Berlin-Heidelberg-New York: Springer 1984
[2] Heyer, H.: Probability measures on locally compact groups. Berlin-Heidelberg-New York: Springer 1977 · Zbl 0376.60002
[3] Roynette, B.: Croissance et mouvements browniens d’un groupe de Lie nilpotent et simplement connexe. Z. Wahrscheinlichkeitstheorie verw. Gebiete32, 133-138 (1975) · Zbl 0312.60036
[4] Siebert, E.: Stetige Halbgruppen von Wahrscheinlichkeitsmassen auf lokalkompakten maximal fastperiodischen Gruppen. Z. Wahrscheinlichkeitstheorie verw. Gebiete25, 269-300 (1973) · Zbl 0252.60003
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