Let $X$ be a Feller process that has strong Feller property. In this paper, we investigate the Feller as well as strong Feller properties of the semigroups generated by multiplicative functionals of $X$ in open sets. Special attention is given to the Feynman-Kac and Girsanov transforms of $X$. Three examples of local Kato class measure that are not of Kato class are given in the last section so that Feller and strong Feller properties hold for corresponding Feynman-Kac semigroup of $X$ in open sets.
"On doubly Feller property." Osaka J. Math. 46 (4) 909 - 930, December 2009.