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 $\mathcal {L}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 of a general symmetric Markov process. An illuminating concrete example for is , where D is a bounded Euclidean domain in is the Laplace operator in D with zero Dirichlet boundary condition and is the fractional Laplacian in D with zero exterior condition. The strong Markov process corresponding to is a Lévy process that is the sum of Brownian motion in and an independent symmetric (2s)-stable process in 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 It? formula, both of which were recently developed in Chen et al. Stochastic calculus for Dirichlet processes (preprint)(2006)]. The research of T.-S. Zhang is supported by the British EPSRC.