  We study the radial part of sub-Riemannian Brownian motion in the context of
totally geodesic foliations. It\^o's formula is proved for the radial processes
associated to Riemannian distances approximating the Riemannian one. We deduce
very general stochastic completeness criteria for the sub-Riemannian Brownian
motion. In the context of Sasakian foliations and H-type groups, one can push
the analysis further, and taking advantage of the recently proved sub-Laplacian
comparison theorems one can compare the radial processes for the sub-Riemannian
distance to one-dimensional model diffusions. As a geometric application, we
prove Cheng's type estimates for the Dirichlet eigenvalues of the
sub-Riemannian metric balls, a result which seems to be new even in the
Heisenberg group.
