Sub-Laplacians on sub-Riemannian manifolds

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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 GordinaLaetsch2014a. This operator is canonical with respect to the horizontal Brownian motion, we are able to define the sub-Laplacian without some a priori choice of measure. The other operator is divω gradₕ for some volume form ω 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.