“…But, by imposing stronger assumptions on measures related to our Feynman-Kac semigroup, we can eliminate the Feller property of the underlying process for Theorems 1.1 and 1.2(1) (Remarks 3.1 and 4.1), which covers one of the results in [28]. In order to deduce our results, we also use the Donsker-Varadhan's large deviation theory as in [25][26][27]29,31]. Our conditions on measures are milder than theirs because of the refinement of the condition for doubly Feller property of the semigroups (see [6]) and the recent developments of the Feynman-Kac formula by continuous additive functionals of zero energy (see [8,4,5]).…”