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 ˜u and ˜u(Xt ) has Fukushima’s decomposition: ˜u(Xt )− ˜u(X0) =Mu t +Nu t , whereMu t is the martingale part and Nu t is the zero energy part. In this paper, we study the strong continuity of the generalized Feynman– Kac semigroup defined by Pu t f (x) = Ex [eNu t f (Xt )], t 0. Two necessary and sufficient conditions for (P u t )t 0 to be strongly continuous are obtained by considering the quadratic form (Qu,D(E)b), where Qu(f, f ) := E(f, f )+E(u, f 2) for f ∈ D(E)b, and the energy measure μ u of u, respectively. An example is also given to show that (P u t )t 0 is strongly continuous when μ u is not a measure of the Kato class but of the Hardy class with the constant δμ u (E) 12 (cf. Definition 4.5). © 2006 Published by Elsevier Inc