  We prove a geometrically meaningful stochastic representation of the
derivative of the heat semigroup on sub-Riemannian manifolds with tranverse
symmetries. This representation is obtained from the study of
Bochner-Weitzenbock type formulas for sub-Laplacians on 1-forms. As a
consequence, we prove new hypoelliptic heat semigroup gradient bounds under
natural global geometric conditions. The results are new even in the case of
the Heisenberg group which is the simplest example of a sub-Riemannian manifold
with transverse symmetries.
