Symmetric Markov processes and large deviation theory

Feynman-Kac泛函的可积性称为可规范性或条件可规范性。条件可规范性与薛定谔算子的次临界性即绿色函数的存在性密切相关。本文给出了广义Schrodinger算子的条件可规范性的一个充要条件，并证明了广义Schrodinger算子的条件可规范性与次临界性的等价性.我们应用这个条件来显示具体的薛定谔算子的次临界性;例如，我们考虑了薛定谔算子，其势是表面的测量。得到了薛定谔算子关于球半径是次临界的一个充分必要条件。为了证明条件可规范性的充分必要条件，我们将Donsker-Varadhan型全大偏差原理推广到有限寿命的对称马氏过程，并将其应用于布朗运动的时变过程.

The integrability of Feynman-Kac functional is called gaugeability or conditional gaugeability. The conditional gaugeability is closely related to the subcriticality of Schrodinger operators, that is, the existence of the Green function. In this study, we obtained a necessary and sufficient condition for conditional gaugeability; moreover we showed the equivalence between conditional gaugeability and subcriticality of generalized Schrodinger operators. We applied this condition to show the subcriticality of concrete Schrodinger operators; for example, we considered the Schrodinger operator whose potential is the surface measure of sphare. We obtained a necessary and sufficient condition the Schrodinger operator being subcritical in terms of the radius of the sphere. To prove the necessary and sufficient condition for conditional gaugeability, we extended the full large deviation principle of Donsker-Varadhan type to symmetric Markov processes with finite lifetime and applied it to a time changed process of the Brownian motion.