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


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


Zbl 0312.60036
Full Text: DOI EuDML


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[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
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