Perturbation of symmetric Markov processes

We present a path-space integral representation of the semigroup associated with the quadratic form obtained by a lower-order perturbation of the L ²-infinitesimal generator {L} of a general symmetric Markov process. An illuminating concrete example for {L} is Δ_D-(-Δ) ^s_D, where D is a bounded Euclidean domain in R ^d, s ε [0, 1], Δ_D is the Laplace operator in D with zero Dirichlet boundary condition and -(-Δ)^s_D is the fractional Laplacian in D with zero exterior condition. The strong Markov process corresponding to L is a Lévy process that is the sum of Brownian motion in R^d and an independent symmetric (2s)-stable process in R^d killed upon exiting the domain D. This probabilistic representation is a combination of Feynman-Kac and Girsanov formulas. Crucial to the development is the use of an extension of Nakao's stochastic integral for zero-energy additive functionals and the associated Itô formula, both of which were recently developed in Chen et al. [Stochastic calculus for Dirichlet processes (preprint)(2006)].