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Asymptotic properties of heat kernels and their applications

热核的渐近性质及其应用

基本信息

批准号：
11640163
负责人：
ICHIHARA Kanji
金额：
$ 2.05万
依托单位：
Nagoya University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2000
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640163/
关键词：
heat kernel harmonic transform curvature pinned process large deviation covering space Gibbs measure Serpinski carpet 負曲率リーマン多様体ラプラシアンスペクトルブラウン運動 Labachevsky空間 pinned process

项目摘要

Ichihara has studied about some relationships between the spectrums of a second order elliptic partial differential operator and the asymptotic properties of the associated heat kernel. First, the exact decay order of the heat kernels in time has been given for a class of complete, simply-connected, negatively curved Riemannian manifolds with asymptotically constant negative sectional curvature. In particular, it has been clarified that there exists an intimate connection between the decay part of polynomial order of the heat kernel and a term of the perturbation of constant negative curvature. Furthermore large deviation principles of the Donsker-Varadhan type have been established for a class of diffusion processes with strong transience property. It is well known that the usual large deviation for a Markov process does not hold when the infimum of the spectrums for its infinitesimal generator ×(-1) is positive. Ichihara has shown that it is still possible to get a nice large deviation principle so far as the pinned process is concerned if an appropriate rate function is chosen.Osada has constructed a Gibbs measure on the path space under the existence of interaction potential and has proved its uniquness in a class of Gibbs measures. Making use of the theory of Dirichlet spaces, a diffusion process having the above Gibbs measure as an invariant probabilty measure has been constructed on the path space. Furthermore Osada has constructed diffusion processes on the Sierpinski Carpet and has given estimates for their transition densities.Chiyonobu has carried out precise estimates for integrations of functionals in an infinite dimensional space, which appear in statistical mechanics.

Ichihara研究了二阶椭圆型偏微分算子的谱与其热核的渐近性质之间的关系。首先，我们给出了一类完备的单连通负曲率黎曼流形的热核在时间上的精确衰减阶，该流形具有渐近常负截面曲率。特别是，它已被澄清，有一个密切的联系之间的衰变部分的多项式阶的热核和一个长期的扰动的常数负曲率。进而对一类具有强瞬态性的扩散过程建立了Donsker-Varadhan型大偏差原理。众所周知，当马氏过程的无穷小生成元×（-1）的谱的下确界为正时，通常的大偏差不成立。Ichihara指出，对于钉扎过程，只要选择适当的速率函数，仍然可以得到一个很好的大偏差原理，Osada在相互作用势存在的情况下构造了一个路径空间上的Gibbs测度，并证明了它在一类Gibbs测度中的唯一性.利用Dirichlet空间理论，在路径空间上构造了一个以上述Gibbs测度为不变概率测度的扩散过程.此外，Osada构造了扩散过程的谢尔宾斯基地毯，并给予估计其过渡密度。千代信进行了精确的估计积分的泛函在一个无限维空间，这出现在统计力学。