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A well-known problem in Malliavin calculus concerns the relation between thedeterminant of the Malliavin matrix of a random vector and the determinant ofits covariance matrix. We give an explicit relation between these twodeterminants for couples of random vectors of multiple integrals. Inparticular, if the multiple integrals are of the same order and this order isat most 4, we prove that two random variables in the same Wiener chaos eitheradmit a joint density, either are proportional and that the result is not truefor random variables in Wiener chaoses of different orders.

The determinant of the Malliavin matrix and the determinant of the covariance matrix for multiple integrals