Asymptotics of heat kernels and principal eigenvalue problems for Laplacians

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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.