In this talk we prove a short time asymptotic expansion of  a hypoelliptic heat kernel on a 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. (This is a joint work with Setsuo Taniguchi.)

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