Functional analytic study on asymptotic properties of Markov processes

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The theory of Dirichlet forms has been developed as a useful tool for studying symmetric Markov processes. The theory of Dirichlet forms is an L-2-theory, and which is a reason why the theory is suitable for treating singular Markov processes. However, the theory of Markov processes is, in a sense, an L-1-theory. To bridge this gap, we prove the L-p-independence of growth bounds of Markov semi-groups under the conditions for the Markov processes to be strong Feller and to be tight. By applying the L-p-independence to time changed Markov processes, we show the exponential integrability of positive continuous additive functionals (PCAF's in short) and the large deviation principle of PCAF's. Moreover, we give a necessary and sufficient condition for heat kernel estimates being stable by perturbation of potential terms.