Formulae for the derivatives of degenerate diffusion semigroups

We consider the diffusion semigroup P_t associated to a class of degenerate elliptic operators A on R^n. This class includes the hypoelliptic Ornstein-Uhlenbeck operator but does not satisfy in general the well-known Hormander condition on commutators for sums of squares of vector fields. We establish probabilistic formulae for the spatial derivatives of P_t f up to the third order. We obtain L^∞-estimates for the derivatives of P_t f and show the existence of a classical bounded solution for the parabolic Cauchy problem involving A and having f ∈ C_b(R^n) as initial datum