Perturbation of symmetric Markov processes

Abstract

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 $$\mathcal {L}$$ of a general symmetric Markov process. An illuminating concrete example for $$\mathcal {L}$$ is $$\Delta_D-(-\Delta)^s_D$$ , where D is a bounded Euclidean domain in $$\mathbb {R}^d, s\in [0, 1], \Delta_D$$ is the Laplace operator in D with zero Dirichlet boundary condition and $$-(-\Delta)^s_D$$ is the fractional Laplacian in D with zero exterior condition. The strong Markov process corresponding to $$\mathcal {L}$$ is a Lévy process that is the sum of Brownian motion in $$\mathbb {R}^d$$ and an independent symmetric (2s)-stable process in $$\mathbb {R}^d$$ 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)].