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Short time full asymptotic expansion of hypoelliptic heat kernel at the cut locus. (English) Zbl 1369.60040

Summary: In this paper 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.

MSC:

60H07 Stochastic calculus of variations and the Malliavin calculus
60H30 Applications of stochastic analysis (to PDEs, etc.)
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60J60 Diffusion processes
58J65 Diffusion processes and stochastic analysis on manifolds
35K08 Heat kernel
41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
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