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SHORT TIME FULL ASYMPTOTIC EXPANSION OF HYPOELLIPTIC HEAT KERNEL AT THE CUT LOCUS

Part of: Approximations and expansions Stochastic analysis Partial differential equations on manifolds; differential operators Parabolic equations and systems

Published online by Cambridge University Press:  10 July 2017

YUZURU INAHAMA  and
SETSUO TANIGUCHI
YUZURU INAHAMA
Affiliation:
Faculty of Mathematics, Kyushu University, Motooka 744, Nishi-ku, Fukuoka, 819-0395, Japan; inahama@math.kyushu-u.ac.jp
SETSUO TANIGUCHI
Affiliation:
Faculty of Arts and Science, Kyushu University, Motooka 744, Nishi-ku, Fukuoka, 819-0395, Japan; se2otngc@artsci.kyushu-u.ac.jp

Abstract

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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 classification

Primary: 60H07: Stochastic calculus of variations and the Malliavin calculus
Secondary: 58J65: Diffusion processes and stochastic analysis on manifolds 35K08: Heat kernel 41A60: Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 5 , 2017 , e16
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2017

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