Strong continuity of generalized Feynman–Kac semigroups: Necessary and sufficient conditions

Let (E,D(E)) be a strongly local, quasi-regular symmetric Dirichlet form on L2(E;m) and ((Xt)t⩾0,(Px)x∈E) the diffusion process associated with (E,D(E)). For u∈D(E)e, u has a quasi-continuous version and has Fukushima's decomposition: , where is the martingale part and is the zero energy part. In this paper, we study the strong continuity of the generalized Feynman–Kac semigroup defined by , t⩾0. Two necessary and sufficient conditions for to be strongly continuous are obtained by considering the quadratic form (Qu,D(E)b), where Qu(f,f):=E(f,f)+E(u,f2) for f∈D(E)b, and the energy measure μ〈u〉 of u, respectively. An example is also given to show that is strongly continuous when μ〈u〉 is not a measure of the Kato class but of the Hardy class with the constant (cf. Definition 4.5).