Stochastic completeness and volume growth in sub-Riemannian manifolds

In this note we generalize A. Grigor'yan's volume test for the stochastic completeness of a Riemannian manifold to a sub-Riemannian setting. As an application of this result, and of a new estimate of the growth of the volume of the metric balls at infinity, we give a different proof of (and extend) a theorem in Baudoin and Garofalo (Arxiv preprint, submitted paper, 2009) stating that when a smooth, complete and connected manifold satisfies the generalized curvature-dimension inequality introduced in that paper, then the manifold turns out to be stochastically complete.