  Given a second order partial differential operator $L$ satisfying the strong
H\"ormander condition with corresponding heat semigroup $P_t$, we give two
different stochastic representations of $dP_t f$ for a bounded smooth function
$f$. We show that the first identity can be used to prove infinite lifetime of
a diffusion of $\frac{1}{2} L$, while the second one is used to find an
explicit pointwise bound for the horizontal gradient on a Carnot group. In both
cases, the underlying idea is to consider the interplay between sub-Riemannian
geometry and connections compatible with this geometry.
