ON THE CONNECTION BETWEEN THE MALLIAVIN COVARIANCE-MATRIX AND HORMANDERS CONDITION

A celebrated theorem of Hörmander gives a sufficient condition for a second order differential operator to be hypoelliptic. For operators with analytic coefficients this condition turns out to be also necessary but this is not true for general smooth coefficients. On the other hand Malliavin conceived a probabilistic approach to the same problem, known as "Malliavin calculus," in which a key role is played by the "Malliavin covariance matrix." The aim of our paper is to give several characterizations of the Malliavin covariance matrix which are equivalent to Hörmander's condition (and consequently imply the hypoellipticity). In this way the distance between Hörmander's condition and the hypoellipticity property is clearly pointed out in probabilistic terms. © 1991.