Advertisement

Krein's spectral theory of strings and generalized diffusion processes

Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 923)

Keywords

Spectral Function Resolvent Operator Generalize Diffusion Tauberian Theorem Order Differential Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    H. Dym and H.P. McKean: Gaussian processes, function theory, and the inverse spectral problem, Academic Press, 1976.Google Scholar
  2. [2]
    K. Itô: Stochastic processes, Iwanami Shoten, 1957 (in Japanese), English translation by Y. Ito, Yale University, 1963.Google Scholar
  3. [3]
    K. Itô and H.P. McKean: Diffusion processes and their sample paths, Springer Verlag, 1965.Google Scholar
  4. [4]
    I.S. Kac: Integral characteristics of the growth of spectral functions for generalized second order boundary problems with conditions at a regular end, Math. USSR Izv. 5 (1971), 161–191 (English transl.).CrossRefzbMATHGoogle Scholar
  5. [5]
    I.S. Kac: Generalization of an asymptotic formula of V.A. Marčenko for spectral functions of a second order boundary value problem, Math. USSR Izv. 7 (1973), 422–436 (English transl.).CrossRefGoogle Scholar
  6. [6]
    I.S. Kac and M.G. Krein: Criteria for the discreteness of the spectrum of a singular string, Izv. Vyss. Učebn. Zaved. Mat. 2(1958), 136–153, (in Russian).MathSciNetGoogle Scholar
  7. [7]
    Y. Kasahara: Spectral theory of generalized second order differential operators and its applications to Markov processes, Japan J. Math. 1(1975), 67–84.MathSciNetzbMATHGoogle Scholar
  8. [8]
    Y. Kasahara: Limit theorems of occupation times for Markov processes, Publ. RIMS, Kyoto Univ. 12(1977), 801–818.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Y. Kasahara: Tauberian theorems of exponential type, J. Math. Kyoto Univ. 18 (1978), 209–219.MathSciNetzbMATHGoogle Scholar
  10. [10]
    Y. Kasahara, S. Kotani and H. Watanabe: On the Green functions of 1-dimensional diffusion processes, Publ. RIMS, Kyoto Univ. 16(1980), 175–188.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    F. Knight: Characterization of the Lévy measures of the inverse local times of gap diffusion, (preprint).Google Scholar
  12. [12]
    S. Kotani: On the inverse problem of M. G. Krein, Sûgaku 27(1973), 266–272 (in Japanese)Google Scholar
  13. [13]
    S. Kotani: On a generalized Sturm-Liouville operator with a singular boundary, J. Math. Kyoto Univ. 15(1975), 423–454.MathSciNetzbMATHGoogle Scholar
  14. [14]
    M.G. Krein: On a generalization of an investigation of Stieltjes, Dokl. Akad. Nauk SSSR 87(1952), 881–884.MathSciNetzbMATHGoogle Scholar
  15. [15]
    M.G. Krein: On some cases of the effective determination of the densities of a non-homogeneous string from its spectral function, Dokl. Acad. Nauk SSSR 93 (1953), 617–620.MathSciNetGoogle Scholar
  16. [16]
    U. Küchler: Some asymptotic properties of the transition densities of one-dimensional quasidiffusions, Publ. RIMS, Kyoto Univ. 16(1980), 245–268.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    H.P. McKean and D.B. Ray: Spectral distribution of a differential operator, Duke Math. J. 29(1962), 281–292.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    T.J. Stieltjes: Recherches sur les fractions continues, Oeuvres Completes, Vol. 2, 1918, 402–566.Google Scholar
  19. [19]
    M. Tomisaki: Comparison theorems in generalized diffusion processes, Mem. Fac. Sci. Kyushu Univ. A. 30 (1976), 248–256MathSciNetzbMATHGoogle Scholar
  20. [20]
    M. Tomisaki: On the asymptotic behaviors of transition probability densities of one-dimensional diffusion processes, Publ. RIMS, Kyoto Univ. 12(1977), 819–837.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    S. Watanabe: On time inversion of one-dimensional diffusion processes, Z. Wahr. verv. Geb. 31(1975), 115–124.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 1982

Authors and Affiliations

  1. 1.Department of MathematicsKyoto UniversityKyotoJapan

Personalised recommendations