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We study criticalities of generalized Schrödinger operators in terms of the Schrödinger forms induced by generalized Feynman-Kac perturbations of symmetric Markov processes, which extend earlier work due to Takeda [26]. The related functional inequalities and analytic characterizations of criticalities of Schrödinger forms are given. As applications, we establish some maximum principles via the analytic characterization.

We establish a large deviation principle for the occupation distribution of a symmetric Markov process normalized by Feynman--Kac functional. The obtained theorem means a large deviation from a ground state, not from an invariant measure.

The uniqueness of invariant measure is one of the most interesting problems in theory of Markov processes. In this paper, we shall prove that the irreducibility in the sense of fine topology implies the uniqueness of invariant probability measures. It is also proven that this irreducibility is strictly weaker than the strong Feller property plus irreducibility in the sense of original topology, which is the usual uniqueness condition.

In this paper, we shall investigate some potential theory for time change of Markov processes. Under weak duality, it is proved that the jumping measure and Feller measure are actually independent of time change, and the jumping measure of a time changed process induced by a PCAF supported on $V$ coincides with the sum of the Feller measure on $V$ and the trace of the original jumping measure on $V$.