Let \(M\) be an \(n\)-dimensional smooth manifold endowed with a complete sub-Riemannian structure and a smooth volume. Relevant results concerning the problem of relating the induced sub-Riemannian distance to the heat kernel were obtained in several papers, e.g. [G. Ben Arous and R. Léandre, Probab. Theory Relat. Fields 90, No. 3, 377–402 (1991; Zbl 0734.60027); T. J. S. Taylor, Pac. J. Math. 136, No. 2, 379–399 (1989; Zbl 0692.35011)]. The main result here relates the heat kernel asymptotics with the so-called hinged energy function on the set of midpoints of all minimizing geodesic connecting \(x\) to \(y\), under the hypothesis that all minimizers connecting \(x\) to \(y\) are strongly normal. This permits to obtain heat kernel asymptotics on the cut locus, filling a gap in the previous literature.
The proof uses the asymptotics of Laplace-type integrals and the relation between the degeneracy of the hinged energy function around the midpoints of the minimal geodesic connecting \(x\) to \(y\) and the conjugacy of the minimal geodesic connecting them. Several applications on Heisenberg group, the nilpotent free case, and the Grushin plane are obtained