Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
Perturbation of symmetric Markov processes
Download PDF
Download PDF
  • Published: 21 March 2007

Perturbation of symmetric Markov processes

  • Z. -Q. Chen1,
  • P. J. Fitzsimmons2,
  • K. Kuwae3 &
  • …
  • T. -S. Zhang4 

Probability Theory and Related Fields volume 140, pages 239–275 (2008)Cite this article

  • 226 Accesses

  • 25 Citations

  • Metrics details

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)].

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Chen Z.-Q. (2003). Analytic characterization of conditional gaugeability for non-local Feynman-Kac transform. J. Funct. Anal. 202: 226–246

    Article  MATH  MathSciNet  Google Scholar 

  2. Chen Z.-Q., Fitzsimmons P.J., Takeda M., Ying J. and Zhang T.-S. (2004). Absolute continuity of symmetric Markov processes. Ann. Probab. 32: 2067–2098

    Article  MATH  MathSciNet  Google Scholar 

  3. Chen, Z.-Q., Fitzsimmons, P.J., Kuwae, K., Zhang, T.-S.: Stochastic calculus for Dirichlet processes (preprint)(2006)

  4. Chen Z.-Q., Ma Z.-M. and Röckner M. (1994). Quasi-homeomorphisms of Dirichlet forms. Nagoya Math. J. 136: 1–15

    MATH  MathSciNet  Google Scholar 

  5. Chen Z.-Q. and Song R. (2003). Conditional gauge theorem for non-local Feynman-Kac transforms. Probab. Theory Relat. Fields 125: 45–72

    Article  MATH  MathSciNet  Google Scholar 

  6. Chen Z.-Q. and Song R. (2003). Drift transforms and Green function estimates for discontinuous processes. J. Funct. Anal. 201: 262–281

    Article  MATH  MathSciNet  Google Scholar 

  7. Chen Z.-Q. and Song R. (2003). Hardy inequality for censored stable processes. Tohoku Math. J. 55: 439–450

    Article  MATH  MathSciNet  Google Scholar 

  8. Chen Z.-Q. and Zhang T.-S. (2002). Girsanov and Feynman-Kac type transformations for symmetric Markov processes. Ann. Inst. H. Poincaré Probab. Stat. 38: 475–505

    Article  MATH  Google Scholar 

  9. Fitzsimmons P.J. (1997). Absolute continuity of symmetric diffusions. Ann. Probab. 25: 230–258

    Article  MATH  MathSciNet  Google Scholar 

  10. Fitzsimmons P.J. and Kuwae K. (2004). Non-symmetric perturbations of symmetric Dirichlet forms. J. Funct. Anal. 208: 140–162

    Article  MATH  MathSciNet  Google Scholar 

  11. Fukushima, M., Ōshima, Y., Takeda, M.: Dirichlet Forms and Symmetric Markov Processes. de Gruyter, Berlin (1994)

  12. He S.W., Wang, J.G., Yan, J.A.: Semimartingale Theory and Stochastic Calculus. Science Press, Beijing New York (1992)

    MATH  Google Scholar 

  13. Kunita, H.: Sub-Markov semi-group in Banach lattices. In: Proceedings of the International Conference on Functional Analysis Relet. Topics, Tokyo, pp. 332–343 (1969)

  14. Kuwae, K.: Maximum principles for subharmonic functions via local semi-Dirichlet forms. Can. J. Math. (to appear) (2006)

  15. Kuwae, K., Takahashi, M.: Kato class functions of Markov processes under ultracontractivity, potential theory in Matsue. In: Adv. Stud. Pure Math., vol. 44, pp. 193–202. Math. Soc. Japan, Tokyo (2006)

  16. Lunt, J., Lyons, T.J., Zhang, T.-S.: Integrability of functionals of Dirichlet processes, probabilistic representations of semigroups, and estimates of heat kernels. J. Funct. Anal. 153, 320–342 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  17. Ma Z.-M., Röckner M. (1992) Introduction to the Theory of (Non-Symmetric) Dirichlet Forms. Springer, Berlin

    MATH  Google Scholar 

  18. Ma Z.-M. and Röckner M. (1995). Markov processes associated with positivity preserving coercive forms. Can. J. Math. 47: 817–840

    MATH  Google Scholar 

  19. Nakao S. (1985). Stochastic calculus for continuous additive functionals of zero energy. Z. Wahrsch. verw. Gebiete. 68: 557–578

    Article  MATH  MathSciNet  Google Scholar 

  20. Sharpe M.(1988) General Theory of Markov Processes. Academic, San Diego

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department ofMathematics, University of Washington, Seattle, WA, 98195, USA

    Z. -Q. Chen

  2. Department of Mathematics, University of California at San Diego, 9500Gilman Drive, La Jolla, CA, 92093-0112, USA

    P. J. Fitzsimmons

  3. Department of Mathematics, Faculty of Education, Kumamoto University, Kumamoto, 860-8555, Japan

    K. Kuwae

  4. School of Mathematics, University of Manchester, Sackville Street, Manchester, M60 1QD, UK

    T. -S. Zhang

Authors
  1. Z. -Q. Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. P. J. Fitzsimmons
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. K. Kuwae
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. T. -S. Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to P. J. Fitzsimmons.

Additional information

In Memory of Professor Martin L. Silverstein.

The research of Z.-Q. Chen is supported in part by NSF Grant DMS-0600206.

The research of P. J. Fitzsimmons is supported by a foundation based on the academic cooperation between Yokohama City University and UCSD.

The research of K. Kuwae is supported by a foundation based on the academic cooperation between Yokohama City University and UCSD, and partially supported by a Grant-in-Aid for Scientific Research (C) No. 16540201 from Japan Society for the Promotion of Science.

The research of T.-S. Zhang is supported by the British EPSRC.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Chen, Z.Q., Fitzsimmons, P.J., Kuwae, K. et al. Perturbation of symmetric Markov processes. Probab. Theory Relat. Fields 140, 239–275 (2008). https://doi.org/10.1007/s00440-007-0065-2

Download citation

  • Received: 16 August 2006

  • Revised: 19 February 2007

  • Published: 21 March 2007

  • Issue Date: January 2008

  • DOI: https://doi.org/10.1007/s00440-007-0065-2

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

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
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature