Small time heat kernel asymptotics at the Riemannian and sub-Riemannian cut locus

For a sub-Riemannian manifold provided with a smooth volume, we relate
the small time asymptotics of the heat kernel at a point $y$ of the
cut locus from $x$ with roughly ``how much'' $y$ is conjugate to $x$.
The result is a refinement of the one of Leandre $4t\log p_t(x,y)\to
-d^2(x,y)$ for $t\to 0$, in which only the leading exponential term is
detected. It are obtained by extending an idea of Molchanov from the
Riemannian to the sub-Riemannian case, and some details we get appear
to be new even in the Riemannian context.
These results permit us to obtain properties of the sub-Riemannian
distance starting from those of the heat kernel and vice versa.