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

摘要

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 is an element of E)) the diffusion process associated with (E, D(E)). For u is an element of D(E)(e), u has a quasi-continuous version (u) over tilde and (u) over tilde (X-t) has Fukushima's decomposition: (u) over tilde (X-t) - (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(Ntu) 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 is an element of D(E)(b), and the energy measure mu([u]) of u, respectively. An example is also given to show that (P-t(u))(t >= 0) is strongly continuous when mu([u]) is not a measure of the Kato class but of the Hardy class with the constant delta(mu[u]) (E) <= 1/2 (cf. Definition 4.5). (C) 2006 Published by Elsevier Inc.