Asymptotics of heat kernels and principal eigenvalue problems for Laplacians

本研究项目的主要目的是系统地研究一类转移概率密度一般随时间指数衰减的可逆马尔可夫过程的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中通常的大偏差结果。在这里，我们关注的是过程中钉扎运动的占据时间分布的大偏差。这种大偏差与相关薛定谔算子的核函数的渐近性有关。我们研究的基本内容是与马尔可夫过程相关的L^2-谱的下界以及相应的广义正特征函数。利用基于上述特征函数的马尔可夫过程的一种调和变换，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关的极限定理：(1)多维方格中具有离散时间参数的可逆周期马氏链，(2)多维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。

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.