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 L-2-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. (C) 2009 Elsevier Masson SAS. All rights reserved.