Uniqueness of embedding of Gaussian probability measures into a continuous convolution semigroup on simply connected nilpotent Lie groups

Daniel Neuenschwander

doi:10.1016/j.crma.2007.10.038

Probability Theory

Uniqueness of embedding of Gaussian probability measures into a continuous convolution semigroup on simply connected nilpotent Lie groups
[Unicité du plongement de mesures de probabilité gaussiennes dans un semigroupe de convolution continu sur des groupes de Lie nilpotents et simplement connexes]

Présenté par : Marc Yor

Daniel Neuenschwander ^1,^2,³

¹ Université de Lausanne, École des hautes études commerciales, Institut de sciences actuarielles, CH-1015 Lausanne, Switzerland
² Universität Bern, Institut für mathematische Statistik und Versicherungslehre, CH-3012 Bern, Switzerland
³ Université de Lyon, Université Claude Bernard Lyon 1, Institut de Science Financière et d'Assurances, 50, Avenue Tony Garnier, F-69007 Lyon, France

Comptes Rendus. Mathématique, Volume 346 (2008) no. 15-16, pp. 887-892.

Résumé
Abstract

Soient ${μ_{t}^{(i)}}_{t ⩾ 0}$ ( $i = 1, 2$ ) des semigroupes de convolution continus sur un groupe de Lie G nilpotent et simplement connexe. Si $μ_{1}^{(1)} = μ_{1}^{(2)}$ et si ${μ_{t}^{(1)}}_{t ⩾ 0}$ est un semigroupe gaussien (au sens que sa distribution génératrice ne consiste que d'une distribution primitive et d'un opérateur différentiel de second ordre), alors $μ_{t}^{(1)} = μ_{t}^{(2)}$ pour tout $t ⩾ 0$ .

Let ${μ_{t}^{(i)}}_{t ⩾ 0}$ ( $i = 1, 2$ ) be continuous convolution semigroups on a simply connected nilpotent Lie group G. Suppose that $μ_{1}^{(1)} = μ_{1}^{(2)}$ and that ${μ_{t}^{(1)}}_{t ⩾ 0}$ is a Gaussian semigroup (in the sense that its generating distribution just consists of a primitive distribution and a second order differential operator). Then $μ_{t}^{(1)} = μ_{t}^{(2)}$ for all $t ⩾ 0$ .

Reçu le : 2006-11-17
Accepté le : 2007-10-22
Publié le : 2008-05-02

DOI : 10.1016/j.crma.2007.10.038

Affiliations des auteurs :

Daniel Neuenschwander ^{1, 2, 3}

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     title = {Uniqueness of embedding of {Gaussian} probability measures into a continuous convolution semigroup on simply connected nilpotent {Lie} groups},
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Daniel Neuenschwander. Uniqueness of embedding of Gaussian probability measures into a continuous convolution semigroup on simply connected nilpotent Lie groups. Comptes Rendus. Mathématique, Volume 346 (2008) no. 15-16, pp. 887-892. doi : 10.1016/j.crma.2007.10.038. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.10.038/

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