Formulae for the derivatives of degenerate diffusion semigroups

Abstract

We consider the diffusion semigroup Pt 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 Hormander condition on commutators for sums of squares of vector fields. We establish probabilistic formulae for the spatial derivatives of Ptf up to the third order. We obtain L∞-estimates for the derivatives of Ptf 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.