Large deviations for symmetric Markov processes and Dirichlet forms

证明了Feynman-Kac泛函的可积性（可规范性）等价于时变过程的主特征值大于1，也等价于Schroedinger算子的次临界性.这一事实说明，时变过程的主特征值准确地度量了测度的大小。利用这一事实，我们得到了三个结果：第一个结果是Schroedinger半群的超压缩性成立当且仅当时变过程的主特征值大于1。第二个结果是分支对称稳定过程中到达闭集的分支数的期望是有限的当且仅当主特征值大于1。最后的结果如下：假设完备黎曼流形上的热核满足整体高斯界，即所谓的Li-Yau估计。则Schroedinger算子的热核也具有整体高斯界，当且仅当主特征值大于1。

We proved that the integrability of Feynman-Kac functionals (gaugeability) is equivalent to that the principal eigenvalue of time-changed process is greater than 1, which is also equivalent to the subcriticality of Schroedinger operators. This fact says that the principal eigenvalue of time-changed process accurately measures the size of measures. Using this fact, we obtained three results : The first result is that the ultracontractiyity of Schroedinger semigroups holds if and only if the princilal eigenvalue of time-changed process is greater than 1. The second result is that the expectation of the number of branches hitting a closed set in a branching symmetric stable process is finite if and only if the princilal eigenvalue is greater than 1. The final result is as foolows : Suppose that the heat kernel on a complete Riemannian manifold satisfies the global Gaussian bounds, so called Li-Yau estimate. Then the heat kernel of the Schroedinger operator also possesses the global Gaussian bounds, if and anly if the princilal eigenvalue is greater than 1.