Perturbation of symmetric Markov processes

• 成果类型: Article
• 署名作者: Chen, Z.-Q.; Fitzsimmons, P. J.; Kuwae, K.; Zhang, T.-S.
• 署名单位: University of California System; University of California San Diego; University of Washington; University of Washington Seattle; Kumamoto University; University of Manchester
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-007-0065-2
• 发表日期: 2008
• 页码: 239-275
• 关键词: feynman-kac transforms; Absolute continuity; dirichlet forms

摘要

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 Delta(D)- (-Delta)(s)(D), where D is a bounded Euclidean domain in R-d, s epsilon [ 0, 1], Delta(D) is the Laplace operator in D with zero Dirichlet boundary condition and -(-Delta)(D)(s) is the fractional Laplacian in D with zero exterior condition. The strong Markov process corresponding to Lis a Levy 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 Ito formula, both of which were recently developed in Chen et al. [Stochastic calculus for Dirichlet processes (preprint)(2006)].