Asymptotics of heat kernels on non-euclidean spaces and spectrum

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An infinite homogenous tree is a typical example of non-euclidean discrete spaces. We have established Donsker-Varadhan's type large deviation for the pinned motions of a radial random walk on the above tree. It has been shown that the corresponding rate function is related to a new Markov chain defined through harmonic transform based on a positive principal eigenfunction for the generator associated with the original random walk. Note that the principal eigenfunction depends only on the structure of the tree. Secondly, there have discussed the same problems for Brownian motions on a class of hyperbolic Riemannian manifolds whose sectional curvature diverges to -∞ at infinity. We have succeeded in showing the uniform large deviation principle for this case. Namely the upper bound is proven to be valid for any closed subset. For the manifold the bottom of the spectrum of the negative Laplacian is discrete and the associated principal eigenfunction decays faster than in an exponential order. Thirdly, the explosion problem for a continuous time, reversible Markov chains on a countably infinite set has been discussed from the viewpoint of Dirichlet space.