On general perturbations of symmetric Markov processes

Let X be a symmetric right process, and let Z = {Z_t, t ≥ 0} be a multiplicative functional of X that is the product of a Girsanov transform, a Girsanov transform under time-reversal and a continuous Feynman-Kac transform. In this paper we derive necessary and sufficient conditions for the strong L²-continuity of the semigroup {T_t, t ≥ 0} given by T_t f (x) = E_x [Z_t f (X_t)], expressed in terms of the quadratic form obtained by perturbing the Dirichlet form of X in the appropriate way. The transformations induced by such Z include all those treated previously in the literature, such as Girsanov transforms, continuous and discontinuous Feynman-Kac transforms, and generalized Feynman-Kac transforms.