Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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THE MEASURE-VALUED DYSON SERIES AND ITS STABILITY THEOREM

  • Ryu, Kun-Sik (Department of Mathematics Han Nam University) ;
  • Im, Man-Kyu (Department of Mathematics Han Nam University)
  • Published : 2006.05.01
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Abstract

In this article, we establish the existence theorem for measure-valued Dyson series and show that it satisfies the Volterra-type integral equation. Furthermore, we prove the stability theorems for measure-valued Dyson series.

Keywords

References

  1. H. Bergstrom, Weak convergence of measures, Academic Press, 1982
  2. P. Billingsley, Convergence of probability measures, John Wiley & Sons, New York, 1968
  3. C. W. Burrill, Measure, integration and probability, McGraw-Hill, New York, 1972
  4. R. H. Cameron and D. A. Storvick, An operator-valued function space integral and a related integral equation, J. Math. Mech. 18 (1968), 517-552
  5. D. L. Cohn, Measure theory, Birkhauser, Boston, 1980
  6. J. Diestel and J. J. Uhl, Vector measures, Mathematical Survey, Amer. Math. Soc. Providence, 1977
  7. N. Dunford and J. T. Schwartz, Linear operators, part I, general theory, Pure and Applied Mathematics, Vol. VII, Wiley Interscience, New York, 1958
  8. P. R. Halmos, Measure theory, Springer-Verlag, New York, 1950
  9. E. Hewitt and K. Stromberg, Real and abstract analysis, Springer-Verlag, New York, 1965
  10. G. W. Johnson and M. L. Lapidus, Generalized Dyson series, generalized Feyn- man diagrams, the Feynman integral and Feynman's operational calculus, Memoirs Amer. Math. Soc. 62 (1986), no. 351, 1-78
  11. G. W. Johnson and M. L. Lapidus, The Feynman integral and Feynman's operational calculus, Oxford Mathematical Monographs, Oxford Univ. Press, 2000
  12. I. Kluvanek, Operator valued measures and perturbations of semi-groups, Arch. Rational Mech. Anal. 81 (1983), no. 2, 161-180 https://doi.org/10.1007/BF00250650
  13. I. Kluvanek and G. Knowles, Vector measures and control systems, Math. Stud ies, no. 20, Amsterdam, North-Holland, 1975
  14. M. L. Lapidus, The Feynman-Kac formula with a Lebesgue-Stieltjes measure and Feynman's operational calculus, Stud. Appl. Math. 76 (1987), no. 2, 93-132 https://doi.org/10.1002/sapm198776293
  15. M. L. Lapidus, Strong product integration of measures and the Feynman-Kac formula with a Lebesgue-Stieltjes measure, Rend. Circ. Math. Palermo (2) Suppl. No. 17 (1987), 271-312
  16. M. L. Lapidus, The Feynman-Kac formula with a Lebesgue-Stieltjes measure: An inte- gral equation in the general case, Integral Equations Operator Theory 12 (1989), no. 2, 163-210 https://doi.org/10.1007/BF01195113
  17. D. R. Lewis, Integration with respect to vector measure, Pacific J. Math. 33 (1970), no. 1, 157-165 https://doi.org/10.2140/pjm.1970.33.157
  18. G. G. Okikiolu, Aspect of the theory of bounded linear operators in $L_p$ space, Academic Press, London, 1971
  19. K. R. Parthasarathy, Probability measures on metric spaces, Academic Press, New York, 1967
  20. W. Rudin, Real and complex analysis, 3rd ed., McGraw-Hill, New York, 1987
  21. K. S. Ryu and M. K. Im, A measure-valued analogue of Wiener measure and the measure-valued Feynman-Kac formula, Trans. Amer. Math. Soc. 354 (2002), no. 12, 4921-4951 https://doi.org/10.1090/S0002-9947-02-03077-5
  22. K. S. Ryu and M. K. Im, An analogue of Wiener measure and its applications, J. Korean Math. Soc. 39 (2002), no. 5, 801-819 https://doi.org/10.4134/JKMS.2002.39.5.801
  23. C. Schwarz, Vector measures of bounded variation, Rev. Roum. Math. Pures. ET Appl Tome XVII (1972), no. 10, 1703-1704
  24. H. G. Tucker, A graduate course in probability, Academic Press, New York, 1967
  25. J. Yeh, Inversion of conditional expectations, Pacific J. Math. 52 (1974), no. 2, 631-640 https://doi.org/10.2140/pjm.1974.52.631
  26. J. Yeh, Stochastic processes and the Wiener integral, Marcel Deckker, New York, 1973

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