On doubly Feller property

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.