On General Perturbations of Symmetric Markov Processes

Z.-Q. Chen, P.J. Fitzsimmons, K. Kuwae, T.-S. Zhang

Let 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[Zt f(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.
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November 12, 2008