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.
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Uwe Franz 1; Nicolas Privault 2; René Schott 3
@article{CRMATH_2003__337_9_609_0, author = {Uwe Franz and Nicolas Privault and Ren\'e Schott}, 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}, }
Uwe Franz; Nicolas Privault; René Schott. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2003.09.014/
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