On general perturbations of symmetric Markov processes

AbstractLet X be a symmetric right process, and let Z={Zt,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 {Tt,t⩾0} given by Ttf(x)=Ex[Ztf(Xt)], 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