Asymptotics of heat kernels on non-euclidean spaces and spectrum

无限齐次树是非欧氏离散空间的典型例子。我们在上面的树上建立了径向随机游动的钉扎运动的Donsker-Varadhan大偏差类型。证明了相应的速率函数与一个新的马尔可夫链有关，该新的马尔可夫链是基于与原始随机游动有关的生成元的正主特征函数的调和变换而定义的。请注意，主特征函数仅取决于树的结构。其次，讨论了截面曲率在无穷远处发散到-∞的一类双曲黎曼流形上的布朗运动。我们成功地给出了这种情况下的一致大偏差原理。也就是说，上界被证明对任何闭子集是有效的。对于流形，负拉普拉斯函数的谱的底部是离散的，并且相关的主特征函数比指数级衰减得更快。第三，从Dirichlet空间的角度讨论了可数无限集上连续时间可逆马氏链的爆炸问题。

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.