Skip to main content
SpringerLink
Log in

Eigenvalues of the Cauchy process on an interval have at most double multiplicity

  • Research Article
  • Published:
Semigroup Forum Aims and scope Submit manuscript

Abstract

We prove that the eigenvalues of the semigroup of the Cauchy process killed upon exiting the interval have at most double multiplicity. In passing we obtain an interesting identity involving Fourier transform.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alessandrini, G.: On Courant’s nodal domain theorem. Forum Math. 10, 521–532 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bañuelos, R., Kulczycki, T.: The Cauchy process and the Steklov problem. J. Funct. Anal. 211(2), 355–423 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bañuelos, R., Kulczycki, T.: Trace estimates for stable processes. Probab. Theory Relat. Fields 142, 313–338 (2008)

    Article  MATH  Google Scholar 

  4. Bañuelos, R., Latała, R., Méndez-Hernández, P.J.: A Brascamp-Lieb-Luttinger type inequality and applications to symmetric stable processes. Proc. Am. Math. Soc. 129, 2997–3008 (2001)

    Article  MATH  Google Scholar 

  5. Bañuelos, R., Kulczycki, T., Méndez-Hernández, P.J.: On the shape of the ground state eigenfunction for stable processes. Potential Anal. 24(3), 205–221 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  6. Blumenthal, R.M., Getoor, R.K.: The asymptotic distribution of the eigenvalues for a class of Markov operators. Pacific J. Math. 9(2), 399–408 (1959)

    MATH  MathSciNet  Google Scholar 

  7. Carlen, E.A., Kusuoka, S., Stroock, D.W.: Upper bounds for symmetric Markov transition functions. Ann. Inst. H. Poincaré Probab. Stat. 23, 245–287 (1987)

    MathSciNet  Google Scholar 

  8. Chavel, I.: Eigenvalues in Riemannian Geometry. Pure and Applied Mathematics, vol. 115. Academic Press, New York (1984)

    MATH  Google Scholar 

  9. Chen, Z.Q., Kumagai, T.: Heat kernel estimates for jump processes of mixed types on metric measure spaces. Probab. Theory Relat. Fields 140, 277–317 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  10. Chen, Z.Q., Song, R.: Intrinsic ultracontractivity, conditional lifetimes and conditional gauge for symmetric stable processes on rough domains. Ill. J. Math. 44(1), 138–160 (2000)

    MATH  MathSciNet  Google Scholar 

  11. Chen, Z.Q., Song, R.: Two sided eigenvalue estimates for subordinate Brownian motion in bounded domains. J. Funct. Anal. 226, 90–113 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  12. Chen, Z.Q., Song, R.: Continuity of eigenvalues of subordinate processes in domains. Math. Z. 252, 71–89 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  13. Chen, Z.Q., Song, R.: Spectral properties of subordinate processes in domains. In: Chen, G.-Q., Hsu, E., Pinsky, M. (eds.) Stochastic Analysis and Partial Differential Equations. Contemporary Mathematics, vol. 429, pp. 77–84. AMS, Providence (2007)

    Google Scholar 

  14. Davies, E.B.: Heat Kernels and Spectral Theory. Cambridge University Press, Cambridge (1989)

    MATH  Google Scholar 

  15. DeBlassie, R.D.: Higher order PDE’s and symmetric stable processes. Probab. Theory Relat. Fields 129, 495–536 (2004)

    MathSciNet  Google Scholar 

  16. DeBlassie, R.D., Méndez-Hernández, P.J.: α-continuity properties of the symmetric α-stable process. Trans. Am. Math. Soc. 359, 2343–2359 (2007)

    Article  MATH  Google Scholar 

  17. Doob, J.L.: A probability approach to the heat equation. Trans. Am. Math. Soc. 80, 216–280 (1955)

    Article  MATH  MathSciNet  Google Scholar 

  18. Dyda, B., Kulczycki, T.: Spectral gap for stable process on convex planar double symmetric domains. Potential Anal. 27(2), 101–132 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  19. Getoor, R.K.: Markov operators and their associated semi-groups. Pac. J. Math. 9, 449–472 (1959)

    MATH  MathSciNet  Google Scholar 

  20. Getoor, R.K.: First passage times for symmetric stable processes in space. Trans. Am. Math. Soc. 101, 75–90 (1961)

    Article  MATH  MathSciNet  Google Scholar 

  21. Hardy, G.: Note on a theorem of Hilbert concerning series of positive terms. Proc. Lond. Math. Soc. 23(2) (1925). Records of Proc. XLV-XLVI

  22. Hunt, R.A.: Some theorems concerning Brownian motion. Trans. Am. Math. Soc. 81, 294–319 (1956)

    Article  MATH  Google Scholar 

  23. Kac, M.: On some connections between probability theory and differential and integral equations. In: Proceedings of the 2nd Berkeley Symposium on Mathematical Statistics and Probability, pp. 189–215. University of California Press, Berkeley

  24. Kakutani, S.: Two-dimensional Brownian motion and harmonic functions. Proc. Imp. Acad. 20(10), 706–714 (1944)

    Article  MATH  MathSciNet  Google Scholar 

  25. Kallenberg, O.: Foundations of Modern Probability. Springer, Berlin (1997)

    MATH  Google Scholar 

  26. Kulczycki, T.: Intrinsic ultracontractivity for symmetric stable processes. Bull. Pol. Acad. Sci. Math. 46(3), 325–334 (1998)

    MATH  MathSciNet  Google Scholar 

  27. Kozlov, V., Kuznetsov, N.G., Motygin, O.: On the two-dimensional sloshing problem. Proc. R. Soc. Lond. A 460, 2587–2603 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  28. Kuznetsov, N., Motygin, O.: The Steklov problem in a half-plane: the dependence of eigenvalues on a piecewise-constant coefficient. J. Math. Sci. 127(6), 2429–2445 (2005)

    Article  MathSciNet  Google Scholar 

  29. Kwaśnicki, M.: Spectral gap estimate for stable processes on arbitrary bounded open sets. Probab. Math. Stat. 28(1), 163–167 (2008)

    MATH  Google Scholar 

  30. Kwaśnicki, M.: Intrinsic ultracontractivity for stable semigroups on unbounded open sets. Potential Anal. 31(1), 57–77 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  31. Song, R., Vondracek, Z.: Potential theory of subordinate killed Brownian motion in a domain. Probab. Theory Relat. Fields 125(4), 578–592 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  32. Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970)

    MATH  Google Scholar 

  33. Troesch, A.B., Troesch, R.: A remark on the sloshing frequencies for a half-space. Z. Angew. Math. Phys. 23(4), 703–711 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  34. Watanabe, S.: On stable processes with boundary conditions. J. Math. Soc. Jpn. 14(2), 170–198 (1962)

    Article  MATH  Google Scholar 

  35. Yosida, K.: Functional Analysis. Springer, Berlin (1980)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics and Computer Science, Wrocław University of Technology, ul. Wybrzeże Wyspiańskiego 27, 50-370, Wrocław, Poland

    Mateusz Kwaśnicki

Authors
  1. Mateusz Kwaśnicki
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mateusz Kwaśnicki.

Additional information

Communicated by Gabriella DiBlasio.

The research was supported by KBN grant 1 P03A 020 28.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kwaśnicki, M. Eigenvalues of the Cauchy process on an interval have at most double multiplicity. Semigroup Forum 79, 183–192 (2009). https://doi.org/10.1007/s00233-009-9166-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00233-009-9166-9

Keywords

Search

Navigation