Non-Gaussian Malliavin calculus on real Lie algebras (Q1763937)

The article introduces a non-commutative Malliavin calculus on the affine algebra which is generated by the matrices \[ X_1=\left( \begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix} \right), \qquad X_2 = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} \] satisfying the relationship \([X_1,X_2]=X_2\). A Malliavin gradient operator \(D_x\) and a Skorokhod integration operator \(\delta\) are defined and an integration by parts formula as well as certain commutation rules are established. The results obtained in this article are related to results obtained by \textit{U. Franz, R. Léandre} and \textit{R. Schott} [Infin. Dimens. Anal. Quantum Probab. Relat. Top. 4, 11--38 (2001; Zbl 1043.81041)], where a non-commutative Malliavin calculus on the Heisenberg-Weyl algebra \(\{p, q, I\}\) with \([p,q]=2iI\) was introduced, generalizing the Gaussian Malliavin calculus to Wigner densities [see also \textit{S. T. Ali, N. M. Atakishiyev, S. M. Chumakov} and \textit{K. B. Wolf}, Ann. Henri Poincaré 1, 685--714 (2000; Zbl 1024.81015)]. This article however aims to treat other probability laws in a more general framework, in particular non-commutative couples of random variables with gamma and continuous binomial marginals.