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
Sharp bounds on the density, Green function and jumping function of subordinate killed BM
Download PDF
Download PDF
  • Published: 02 January 2004

Sharp bounds on the density, Green function and jumping function of subordinate killed BM

  • Renming Song1 

Probability Theory and Related Fields volume 128, pages 606–628 (2004)Cite this article

  • 203 Accesses

  • 27 Citations

  • Metrics details

Abstract.

Subordination of a killed Brownian motion in a domain D⊂ℝd via an α/2-stable subordinator gives rise to a process Z t whose infinitesimal generator is −(−Δ| D )α/2, the fractional power of the negative Dirichlet Laplacian. In this paper we establish upper and lower estimates for the density, Green function and jumping function of Z t when D is either a bounded C 1,1 domain or an exterior C 1,1 domain. Our estimates are sharp in the sense that the upper and lower estimates differ only by a multiplicative constant.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Blumenthal, R.M., Getoor, R.K.: Some theorems on stable processes. Trans. Am. Math. Soc. 95, 263–273 (1960)

    MATH  Google Scholar 

  2. Blumenthal, R.M., Getoor, R.K.: Markov processes and potential theory. Academic Press, New York, 1968

  3. Bouleau, N.: Quelques résultats probabilistes sur la subordination au sens de Bochner. In: Seminar on Potential Theory, Paris, No. 7, Lecture Notes in Math., 1061, Springer, Berlin, 1984, pp. 54–81

  4. Chen, Z.-Q., Song, R.: Estimates on Green functions and Poisson kernels for symmetric stable processes. Math. Ann. 312, 465–501 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chen,Z.-Q., Song, R.: General gauge and conditional gauge theorems. Ann. Probab. 30, 1313–1339 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chung, K.L., Zhao, Z.: From Brownian motion to Schrödinger equations. Springer, Berlin, 1995

  7. Davies, E.B.: Heat kernels and spectral theory. Cambridge University Press, Cambridge, 1989

  8. Fabes, E.B., Garofalo, N., Salsa, S.: A backward Harnack inequality and Fatou theorem for nonnegative solutions of parabolic equations. Illinois J. Math. 30, 536–565 (1986)

    MathSciNet  MATH  Google Scholar 

  9. Farkas, W., Jacob, N.: Sobolev spaces on non-smooth domains and Dirichlet forms related to subordinate reflecting diffusions. Math. Nachr. 224, 75–104 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  10. Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet forms and symmetric Markov processes. Walter De Gruyter, Berlin, 1994

  11. Glover, J., Rao, M., Šikić, H., Song, R.: Γ-potentials. In: Classical and modern potential theory and applications (Chateau de Bonas, 1993), Kluwer Acad. Publ., Dordrecht, 1994, pp. 217–232

  12. Ikeda, N., Watanabe, S.: On some relations between the harmonic measure and the Lévy measure for a certain class of Markov processes. J. Math. Kyoto Univ. 2, 79–95 (1962)

    MATH  Google Scholar 

  13. Jacob, N., Schilling, R.: Some Dirichlet spaces obtained by subordinate reflected diffusions. Rev. Mat. Iberoamericana 15, 59–91 (1999)

    MathSciNet  MATH  Google Scholar 

  14. Kulczycki, T.: Properties of Green function of symmetric stable processes. Probab. Math. Statist. 17, 339–364 (1997)

    MATH  Google Scholar 

  15. Miyake, A.: The subordination of Lévy system for Markov processes. Proc. Japan Acad. 45, 601–604 (1969)

    MATH  Google Scholar 

  16. Pazy, A.: Semigroups of linear operators and applications to partial differential equations. Springer-Verlag, New York, 1983

  17. Schilling, R.L.: On the domain of the generator of a subordinate semigroup. In: J. Král et al. (eds.), Potential theory– ICPT 94. Proceedings Intl. Conf. Potential Theory, Kouty (CR), 1994 de Gruyter. 1996, pp. 449–462

  18. Song, R., Vondracek, Z.: Potential theory of subordinate killed Brownian motion in a domain. Probab. Theory Relat. Fields. 135, 578–592 (2003)

    Google Scholar 

  19. Yosida, K.: Functional analysis. 6th Edition, Springer-Verlag, Berlin, 1980

  20. Zhang, Q.S.: The boundary behavior of heat kernels of Dirichlet Laplacians. J. Diff. Eqs. 182, 416–430 (2002)

    Article  MATH  Google Scholar 

  21. Zhang, Q.S.: The global behavior of heat kernels in exterior domains. J. Funct. Anal. 200, 160–176 (2003)

    Article  MATH  Google Scholar 

  22. Zolotarev, V.M.: One-dimensional stable distributions. Am. Math. Soc. Providence, RI, 1986

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Illinois, Urbana, IL, 61801, USA

    Renming Song

Authors
  1. Renming Song
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Renming Song.

Additional information

Mathematics Subject Classification (2000): Primary 60J45, Secondary 60J75, 31C25

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Song, R. Sharp bounds on the density, Green function and jumping function of subordinate killed BM. Probab. Theory Relat. Fields 128, 606–628 (2004). https://doi.org/10.1007/s00440-003-0316-9

Download citation

  • Received: 17 February 2003

  • Revised: 23 October 2003

  • Published: 02 January 2004

  • Issue Date: April 2004

  • DOI: https://doi.org/10.1007/s00440-003-0316-9

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

  •  Killed Brownian motions
  • Stable processes
  • Subordination
  • Fractional Laplacian
  • Transition density
  • Green function
  • Jumping function
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