[Régularité de densités de Wigner sur l'algèbre affine]
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.
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.
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@article{CRMATH_2003__337_9_609_0,
author = {Franz, Uwe and Privault, Nicolas and Schott, Ren\'e},
title = {Smoothness of {Wigner} densities on the affine algebra},
journal = {Comptes Rendus. Math\'ematique},
pages = {609--614},
publisher = {Elsevier},
volume = {337},
number = {9},
year = {2003},
doi = {10.1016/j.crma.2003.09.014},
language = {en},
url = {http://archive.numdam.org/articles/10.1016/j.crma.2003.09.014/}
}
TY - JOUR AU - Franz, Uwe AU - Privault, Nicolas AU - Schott, René TI - Smoothness of Wigner densities on the affine algebra JO - Comptes Rendus. Mathématique PY - 2003 SP - 609 EP - 614 VL - 337 IS - 9 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.crma.2003.09.014/ DO - 10.1016/j.crma.2003.09.014 LA - en ID - CRMATH_2003__337_9_609_0 ER -
%0 Journal Article %A Franz, Uwe %A Privault, Nicolas %A Schott, René %T Smoothness of Wigner densities on the affine algebra %J Comptes Rendus. Mathématique %D 2003 %P 609-614 %V 337 %N 9 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.crma.2003.09.014/ %R 10.1016/j.crma.2003.09.014 %G en %F CRMATH_2003__337_9_609_0
Franz, Uwe; Privault, Nicolas; Schott, René. Smoothness of Wigner densities on the affine algebra. Comptes Rendus. Mathématique, Tome 337 (2003) no. 9, pp. 609-614. doi : 10.1016/j.crma.2003.09.014. http://archive.numdam.org/articles/10.1016/j.crma.2003.09.014/
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