On Girsanov and generalized Feynman-Kac transformations for symmetric processes

Let X be a Markov process, which is assumed to be associated with a symmetric Dirichlet form (epsilon, D(epsilon)). For u is an element of D(epsilon)(e), the extended Dirichlet space, we have the classical Fukushima's decomposition: (u) over tilde (X-t) - (u) over tilde (X-0) = M-t(u) + N-t(u), where (u) over tilde is a quasi-continuous version of u, M-t(u) the martingale part and N-t(u) the zero energy part. In this paper, we investigate two important transformations for X, the Girsanov transform induced by M-t(u) and the generalized Feynman-Kac transform induced by N-t(u). For the Girsanov transform, we present necessary and sufficient conditions for which to induce a positive supermartingale and hence to determine another Matkov process (X) over cap. Moreover, we characterize the symmetric Dirichlet form associated with the Girsanov transformed process (X) over cap. For the generalized Feynman-Kac transform, we give a necessary and sufficient condition for the generalized Feynman-Kac semigroup to be strongly continuous.