Formulae for the derivatives of degenerate diffusion semigroups

Abstract.

We consider the diffusion semigroup P _t associated to a class of degenerate elliptic operators $\mathcal{A}\; \text{on}\; \mathbb{R}^n$. This class includes the hypoelliptic Ornstein-Uhlenbeck operator but does not satisfy in general the well-known Hörmander 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 $\mathcal{A}$ and having $f\in C_b(\mathbb{R}^n)$ as initial datum.