SHORT TIME FULL ASYMPTOTIC EXPANSION OF HYPOELLIPTIC HEAT KERNEL AT THE CUT LOCUS
SHORT TIME FULL ASYMPTOTIC EXPANSION OF HYPOELLIPTIC HEAT KERNEL AT THE CUT LOCUS
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DOI：
10.1017/fms.2017.14
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发表时间：
2016-03
期刊：
影响因子：
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通讯作者：
Y. Inahama;S. Taniguchi
中科院分区：
文献类型：
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作者：
Y. Inahama;S. Taniguchi
在欧氏空间和紧流形上证明了一个准椭圆热核的短时间渐近展开式。我们研究了“切轨迹”的情况，即连接两个点的能量最小化路径不是一个有限集，而是一个紧流形。在温和的假设下，我们得到了热核在任意阶的渐近展开式。我们的方法是概率性的，将热核看作是准椭圆扩散过程的密度，并将其作为相应的随机微分方程的唯一解来实现。我们的主要工具是S. Watanabe的分布Malliavin演算和T. Lyons的粗糙路径理论。
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.