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

Abstract

Let ( E , D ( E ) ) be a strongly local, quasi-regular symmetric Dirichlet form on L 2 ( E ; m ) and ( ( X t ) t ⩾ 0 , ( P x ) x ∈ E ) the diffusion process associated with ( E , D ( E ) ) . For u ∈ D ( E ) e , u has a quasi-continuous version u ˜ and u ˜ ( X t ) has Fukushima's decomposition: u ˜ ( X t ) − u ˜ ( X 0 ) = M t u + N t u , where M t u is the martingale part and N t u is the zero energy part. In this paper, we study the strong continuity of the generalized Feynman–Kac semigroup defined by P t u f ( x ) = E x [ e N t u f ( X t ) ] , t ⩾ 0 . Two necessary and sufficient conditions for ( P t u ) t ⩾ 0 to be strongly continuous are obtained by considering the quadratic form ( Q u , D ( E ) b ) , where Q u ( 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 t u ) t ⩾ 0 is strongly continuous when μ 〈 u 〉 is not a measure of the Kato class but of the Hardy class with the constant δ μ 〈 u 〉 ( E ) ⩽ 1 2 (cf. Definition 4.5).