Symmetric Markov processes and Dirichlet forms

本研究的目标是通过使用Dirichlet表格理论来研究对称马尔可夫工艺。对称马尔可夫处理器是Donsker-Varadhan型大偏差理论中的一个特殊类，其意义是，大偏差原则的速率函数是由相关Dirichlet形式赋予的。在1984年， Fukushima和我指出，即使是一些超级martingale多功能功能的对称马尔可夫过程也可以被转化为ergodic过程，即使是对称马尔可夫过程被利用或在内部被杀死。作为一个结果， Donsker-Varadhan型大型调试原则可以用有限的寿命扩展到对称马尔可夫过程。在这项研究中，我发现了一个新的可行条件来容纳上层估计，不仅是紧凑套，而且也是封闭套。在事实上，我展示了如果马尔可夫过程的充分大偏差原则，那么马尔可夫过程就能如此迅速地利用1个身份函数来消失在无限空间中的连续函数。作为这一结果的综述，我展示了L-D1 p-D1-对称马尔可夫半组的光谱半径的独立性。我已经应用了它来为Feynman-Kac功能的完整性提供必要和有效的条件。这也给出了我们对施罗德运算符是间接还是非临界的标准，我们进一步扩展了使用Feynman-Kac功能和考虑到Feynman-Kac semigroups的马尔可夫过程的大偏差原则。

The objective of this study is to investigate symmetric Markov processes by using Dirichlet form theory. Symmetric Markov processes are a special class in Donsker-Varadhan type large deviation theory in the sense that the rate functions of large deviation principle are given by the associated Dirichlet forms. In 1984, Fukushima and I showed that symmetric Markov processes can be transformed to ergodic processes by some supermartingale multiplicative functionals even if a symmetric Markov process is explosive or has the killing inside. As a result, Donsker-Varadhan type large deviation principle could be extended to symmetric Markov processes with finite lifetime. In this study, I found a new sufficient condition for the upper estimate holding for not only compact sets but also for closed sets. In fact, I showed that the full large deviation principle holds if the Markov process explodes so fast that the 1-resolvent of the identity function belongs to the space of continuous functions vanishing at infinity. As a corollary of this result, I showed LィイD1pィエD1-independence of the spectral radius of symmetric Markov semigroups. And I applied it to obtain a necessary and sufficient condition for the integrability of Feynman-Kac functionals. This result also gives us an criterion whether a Schrodinger operators is subcritical or not.We further extended the large deviation principle to Markov processes with Feynman-Kac functional, and consider asymptotic properties of Feynman-Kac semigroups.