Albanese maps and off diagonal long time asymptotics for the heat kernel
Albanese maps and off diagonal long time asymptotics for the heat kernel
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DOI：
10.1007/s002200050033
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发表时间：
2000-02-01
影响因子：
2.4
通讯作者：
Sunada, T
中科院分区：
文献类型：
--
作者：
Kotani, M;Sunada, T
我们讨论了非紧黎曼流形上热核的长时间渐近行为，该流形允许具有紧商的阿贝尔等距群的间断自由作用。通过对扭曲拉普拉斯函数的特征值和特征函数的摄动，建立了热核的局部中心极限定理和热核随时间参数变化的渐近级数展开式。我们的思想和技巧在一定程度上类似于周期薛定谔算子的Floque-Bloch理论，对于渐近展开，我们谨慎地使用了经典的Laplace方法。在讨论过程中，我们观察到与阿贝尔群行动有关的阿尔巴内斯映射的概念与渐近性密切相关。类似的想法也适用于格图上随机游动转移概率的渐近。本文得到的结果改进了我们以前的结果[4]。在渐近性中，与格图的标准实现有关的欧几里德距离，我们称之为艾博年距离，起着至关重要的作用。
We discuss long time asymptotic behaviors of the heat kernel on a noncompact Riemannian manifold which admits a discontinuous free action of an abelian isometry group with a compact quotient. A local central limit theorem and the asymptotic power series expansion for the heat kernel as the time parameter goes to infinity are established by employing perturbation arguments on eigenvalues and eigenfunctions of twisted Laplacians. Our ideas and techniques are motivated partly by analogy with Floque-Bloch theory on periodic Schrodinger operators, For the asymptotic expansion, we make careful use of the classical Laplace method. In the course of a discussion, we observe that the notion of Albanese maps associated with the abelian group action is closely related to the asymptotics. A similar idea is available for asymptotics of the transition probability of a random walk on a lattice graph. The results obtained in the present paper refine our previous ones [4]. In the asymptotics, the Euclidean distance associated with the standard realization of the lattice graph, which we call the Albanese distance, plays a crucial role.