Girsanov and Feynman-KAC type transformations for symmetric Markov processes

Studied in this paper is the transformation of an arbitrary symmetric Markov process X by multiplicative functionals which are the exponential of continuous additive functionals of X having zero quadratic variations. We characterize the transformed semigroups by their associated quadratic forms. This is done by first identifying the symmetric Markov process under Girsanov transform, which may be of independent interest, and then applying Feynman-Kac transform to the Girsanov transformed process. Stochastic analysis for discontinuous martingales is used in our approach.