Probability Theory
Smoothness of Wigner densities on the affine algebra
Comptes Rendus. Mathématique, Volume 337 (2003) no. 9, pp. 609-614.

The non-commutative Malliavin calculus on the Heisenberg–Weyl algebra (see (i) C. R. Acad. Sci. Paris, Sér. I 328 (11) (1999) 1061–1066, (ii) Infin. Dimens. Anal. Quantum Probab. Relat. Top. 4 (1) (2001) 11–38) is extended to the affine algebra. A differential calculus is established, which generalizes the corresponding commutative integration by parts formulas. As an application we obtain sufficient conditions for the smoothness of Wigner type laws of non-commutative random variables with gamma and continuous binomial marginals.

Le calcul de Malliavin non-commutatif sur l'algèbre de Heisenberg–Weyl (voir (i) C. R. Acad. Sci. Paris, Sér. I 328 (11) (1999) 1061–1066, (ii) Infin. Dimens. Anal. Quantum Probab. Relat. Top. 4 (1) (2001) 11–38) est étendu à l'algèbre affine. Un calcul différentiel non-commutatif qui généralise les formules d'intégration par parties classiques est établi. Comme application nous obtenons des conditions suffisantes pour la régularité de lois de Wigner pour des variables aléatoires non-commutatives de lois marginales gamma et binomiale continue.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2003.09.014
Franz, Uwe 1; Privault, Nicolas 2; Schott, René 3

1 Institut für Mathematik und Informatik, Ernst-Moritz-Arndt-Universität Greifswald, Jahnstraße 15a, 17487 Greifswald, Germany
2 Département de mathématiques, Université de La Rochelle, 17042 La Rochelle, France
3 Institut Elie Cartan and LORIA, BP 239, Université H. Poincaré-Nancy I, 54506 Vandœuvre-lès-Nancy, France
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Franz, Uwe; Privault, Nicolas; Schott, René. Smoothness of Wigner densities on the affine algebra. Comptes Rendus. Mathématique, Volume 337 (2003) no. 9, pp. 609-614. doi : 10.1016/j.crma.2003.09.014. http://www.numdam.org/articles/10.1016/j.crma.2003.09.014/

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