Symmetric Markov processes and Dirichlet forms

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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.