  We show how Molchanov's method provides a systematic approach to determining
the small-time asymptotics of the heat kernel on a sub-Riemannian manifold away
from any abnormal minimizers. The expansion is closely connected to the
structure of the minimizing geodesics between two points. If the normal form of
the exponential map at the minimal geodesics between two points is sufficiently
explicit, a complete asymptotic expansion of the heat kernel can, in principle,
be given. (But one can also exhibit metrics for which the exponential map is
quite degenerate, so that even giving the leading term of the expansion seems
doubtful.) In a different direction, we have uniform bounds on the heat kernel
and its derivatives over any compact set with no abnormals, in an inherently
local way.
  The method extends naturally to logarithmic derivatives of the heat kernel,
which are closely related to the law of large numbers for the corresponding
bridge process. This allows for a general treatment of the small-time behavior
of the bridge process on the cut locus, as well as the determination of the
limiting measure of the bridge process if the normal form of the exponential
map at the minimal geodesics between two points is sufficiently explicit.
Further, we give an expression for the leading term of the logarithmic
derivative of the heat kernel, of any order, as a cumulant of geometrically
natural random variables and a characterization of the cut locus in terms of
the blow-up of the logarithmic second derivative of the heat kernel. This
method also provides uniform, local estimates of the logarithmic derivatives.
