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.