Sub-Laplacians on Sub-Riemannian Manifolds

We consider different sub-Laplacians on a sub-Riemannian manifold M. Namely, we compare different natural choices for such operators, and give conditions under which they coincide. One of these operators is a sub-Laplacian we constructed previously in Gordina and Laetsch (Trans. Amer. Math. Soc., 2015). This operator is canonical with respect to the horizontal Brownian motion; we are able to define this sub-Laplacian without some a priori choice of measure. The other operator is div(omega) grad(H) for some volume form omega on M. We illustrate our results by examples of three Lie groups equipped with a sub-Riemannian structure: SU(2), the Heisenberg group and the affine group.