  In this paper we prove a short time asymptotic expansion of a hypoelliptic
heat kernel on an Euclidean space and a compact manifold. We study the "cut
locus" case, namely, the case where energy-minimizing paths which join the two
points under consideration form not a finite set, but a compact manifold. Under
mild assumptions we obtain an asymptotic expansion of the heat kernel up to any
order. Our approach is probabilistic and the heat kernel is regarded as the
density of the law of a hypoelliptic diffusion process, which is realized as a
unique solution of the corresponding stochastic differential equation. Our main
tools are S. Watanabe's distributional Malliavin calculus and T. Lyons' rough
path theory.
