  We consider the kernel of a hypoelliptic diffusion beyond the case of
sub-ellipticity or polynomial coefficients. We get a full asymptotic expansion
for small times, based on a Duhamel-type comparison with an approximate
polynomial kernel. As in the sub-elliptic case, some change of scale based on
the geometry of some Lie brackets yields a non-trivial limit for the kernel as
time goes to zero. Remarkably, a different scale is needed to observe a
non-trivial large deviation principle.
