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Probability Theory and Related Fields

, Volume 140, Issue 1–2, pp 239–275 | Cite as

Perturbation of symmetric Markov processes

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

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)].

Keywords

Perturbation Symmetric Markov process Time reversal Girsanov transform Feynman-Kac transform Stochastic integral for Dirichlet processes Martingale Revuz measure Dual predictable projection 

Mathematics Subject Classification (2000)

Primary: 31C25 Secondary: 60J57 Secondary: 60J55 Secondary: 60H05 

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Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  • Z. -Q. Chen
    • 1
  • P. J. Fitzsimmons
    • 2
  • K. Kuwae
    • 3
  • T. -S. Zhang
    • 4
  1. 1.Department ofMathematicsUniversity of WashingtonSeattleUSA
  2. 2.Department of MathematicsUniversity of California at San DiegoLa JollaUSA
  3. 3.Department of Mathematics, Faculty of EducationKumamoto UniversityKumamotoJapan
  4. 4.School of MathematicsUniversity of ManchesterManchesterUK

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