Asymptotics of heat kernels and principal eigenvalue problems for Laplacians

热核的渐近性和拉普拉斯算子的主特征值问题

• 批准号: 13640165
• 负责人: ICHIHARA Kanji
• 金额: $ 1.98万
• 依托单位: Kansai University (2002)Nagoya University (2001)
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640165/
• 关键词: heat kernel; Markov process; large deviation; spectrum; principal eigenfunction; hyperbolic space; tree; Brownian motion; ディリクレ形式; マルコフ連鎖; 固定端過程; 調和変換; 主固有値; 被覆空間; 離散群; 対称空間

The main purpose in this research project is a systematic investigation of the Donsker-Varadhan type large deviation for a class of reversible Markov processes whose transition probability densities generally decay exponentially in time. However the Markov processes in this class possesses a strong transience property. Therefore it can not be anymore expected to prove the usual large deviation results for the processes. Here we are concerned with the large deviation of the occupation time distribution for the pinned motions of the processes. Such type of large deviations are relevant to the asymptotics of the kernel functions of the associated Schrodinger operators. The essential ingredients in our research are the lower bound of the L^2-spectrum associated with the Markov process and the corresponding generalized positive eigenfunction. Making use of a kind of harmonic transform of the Markov process based on the above eigenfunction, a rate function suitable to the present case is introduced. We have established large deviation principles and related limit theorems for the following processes:(1) Reversible, periodic Markov chains with discrete time parameter in the multidimensional square lattices,(2) Reversible, periodic Markov chains with continuous time parameter in the multidimensional square lattices,(3) Brownian motions in hyperbolic spaces.(4) Radial random walks in homogeneous trees.

本文的主要目的是系统地研究一类转移概率密度随时间指数衰减的可逆Markov过程的Donsker-Varadhan型大偏差。然而，这类马尔可夫过程具有很强的瞬时性。因此，不能再期望证明过程的通常大偏差结果。在这里，我们关注的是大偏差的占领时间分布的过程的固定运动。这类大偏差与相应薛定谔算子的核函数的渐近性有关。我们研究的基本要素是与马尔可夫过程相关的L^2谱的下界和相应的广义正本征函数。利用马尔可夫过程的一种调和变换，在上述特征函数的基础上，引入了一种适合于这种情况的速率函数。我们建立了下列过程的大偏差原理和相应的极限定理：（1）多维正方格中具有离散时间参数的可逆周期马氏链，（2）多维正方格中具有连续时间参数的可逆周期马氏链，（3）双曲空间中的布朗运动. (4)齐次树中的径向随机游动。

项目成果

福島正俊: "On Sobolev and capacitary inequalities for contractive Besov spaces over d-sets"Potential Analysis. 18. 59-77 (2003)

Masatoshi Fukushima：“关于 d 集上收缩贝索夫空间的索博列夫和电容不等式”潜在分析 18. 59-77 (2003)。

H. Osada: "Harnack inequalities for exotic Brownian motions"Kyushu J. Math.. 56. 363-380 (2002)

H. Osada：“奇异布朗运动的 Harnack 不等式”Kyushu J. Math.. 56. 363-380 (2002)

市原完治: "Long Time Asymptotic Properties of Heat Kernels on Negatively Curved Riemannian Manifolds"Infinite Dimensional Analysis, Quantum Probability and Related Topics. 4. 377-400 (2001)

Kanji Ichihara：“负弯曲黎曼流形上热核的长期渐近性质”无限维分析、量子概率和相关主题 4. 377-400 (2001)。

市原完治: "Birth and death processes in randomly fluctuating environments"Nagoya Math.J.. 166. 93-115 (2002)

市原宽治：“随机波动环境中的生死过程”Nagoya Math.J.. 166. 93-115 (2002)

K. Ichihara: "Birth and death processes in randomly fluctuating environments"Nagoya Math. J.. 166. 93-115 (2002)

K. Ichihara：“随机波动环境中的出生和死亡过程”名古屋数学。