Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

GENERALIZED ANALYTIC FEYNMAN INTEGRAL VIA FUNCTION SPACE INTEGRAL OF BOUNDED CYLINDER FUNCTIONALS

  • Received : 2009.08.01
  • Published : 2011.05.31
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we use a generalized Brownian motion to define a generalized analytic Feynman integral. We then obtain some results for the generalized analytic Feynman integral of bounded cylinder functionals of the form F(x) = $\hat{v}$(($g_1,x)^{\sim}$,..., $(g_n,x)^{\sim}$) defined on a very general function space $C_{a,b}$[0,T]. We also present a change of scale formula for function space integrals of such cylinder functionals.

Keywords

References

  1. R. H. Cameron and D. A. Storvick, An $L_{2}$ analytic Fourier-Feynman transform, Michigan Math. J. 23 (1976), no. 1, 1-30. https://doi.org/10.1307/mmj/1029001617
  2. R. H. Cameron and D. A. Storvick, Relationships between the Wiener integral and the analytic Feynman integral, Rend. Circ. Mat. Palermo (2) Suppl. No. 17 (1987), 117-133.
  3. R. H. Cameron and D. A. Storvick, Change of scale formulas for Wiener integral, Rend. Circ. Mat. Palermo (2) Suppl. No. 17 (1987), 105-115.
  4. K. S. Chang, G. W. Johnson, and D. L. Skoug, Necessary and sufficient conditions for the Fresnel integrability of certain classes of functions, J. Korean Math. Soc. 21 (1984), no. 1, 21-29.
  5. K. S. Chang, G. W. Johnson, and D. L. Skoug, Necessary and sufficient conditions for membership in the Banach algebra S for certain classes of functions, Rend. Circ. Mat. Palermo (2) Suppl. No. 17 (1987), 153-171.
  6. S. J. Chang and D. M. Chung, Conditional function space integrals with applications, Rocky Mountain J. Math. 26 (1996), no. 1, 37-62. https://doi.org/10.1216/rmjm/1181072102
  7. S. J. Chang, J. G. Choi, and D. Skoug, Integration by parts formulas involving generalized Fourier-Feynman transforms on function space, Trans. Amer. Math. Soc. 355 (2003), no. 7, 2925-2948. https://doi.org/10.1090/S0002-9947-03-03256-2
  8. S. J. Chang, J. G. Choi, and D. Skoug, Evaluation formulas for conditional function space integrals. I, Stoch. Anal. Appl. 25 (2007), no. 1, 141-168. https://doi.org/10.1080/07362990601052185
  9. S. J. Chang and D. Skoug, Generalized Fourier-Feynman transforms and a rst variation on function space, Integral Transforms Spec. Funct. 14 (2003), no. 5, 375-393. https://doi.org/10.1080/1065246031000074425
  10. B. S. Kim and T. S. Kim, Change of scale formulas for Wiener integral over paths in abstract Wiener space, Commun. Korean Math. Soc. 21 (2006), no. 1, 75-88. https://doi.org/10.4134/CKMS.2006.21.1.075
  11. Y. S. Kim, Analytic Feynman integrals, Fourier-Feynman transforms and change of scale formula for Wiener integrals over paths on abstract Wiener spaces, Integral Transforms Spec. Funct. 16 (2005), no. 4, 323-335. https://doi.org/10.1080/10652460512331342552
  12. Y. S. Kim, J. M. Ahn, K. S. Chang, and I. Yoo, A change of scale formula for Wiener integrals on the product abstract Wiener spaces, J. Korean Math. Soc. 33 (1996), no. 2, 269-282.
  13. H. L. Royden, Real Analysis, Third edition, Macmillan, 1988.
  14. J. Yeh, Singularity of Gaussian measures on function spaces induced by Brownian motion processes with non-stationary increments, Illinois J. Math. 15 (1971), 37-46.
  15. J. Yeh, Stochastic Processes and the Wiener Integral, Marcel Dekker, Inc., New York, 1973.
  16. I. Yoo, Sequential Yeh-Feynman integrals, Doctoral Thesis, Yonsei University, 1987.
  17. I. Yoo and D. Skoug, A change of scale formula for Wiener integrals on abstract Wiener spaces, Internat. J. Math. Math. Sci. 17 (1994), no. 2, 239-247. https://doi.org/10.1155/S0161171294000359
  18. I. Yoo and D. Skoug, A change of scale formula for Wiener integrals on abstract Wiener spaces. II, J. Korean Math. Soc. 31 (1994), no. 1, 115-129.
  19. I. Yoo, T. S. Song, and B. S. Kim, A change of scale formula for Wiener integrals of unbounded functions. II, Commun. Korean Math. Soc. 21 (2006), no. 1, 117-133. https://doi.org/10.4134/CKMS.2006.21.1.117
  20. I. Yoo, T. S. Song, B. S. Kim, and K. S. Chang, A change of scale formula for Wiener integrals of unbounded functions, Rocky Mountain J. Math. 34 (2004), no. 1, 371-389. https://doi.org/10.1216/rmjm/1181069911
  21. I. Yoo and G. J. Yoon, Change of scale formulas for Yeh-Wiener integrals, Commun. Korean Math. Soc. 6 (1991), no. 1, 19-26.

Cited by

  1. Change of path formula on the function space with applications vol.25, pp.3, 2014, https://doi.org/10.1016/j.indag.2014.02.004
  2. A Modified Analytic Function Space Feynman Integral and Its Applications vol.2014, 2014, https://doi.org/10.1155/2014/671960
  3. New expressions of the modified generalized integral transform via the translation theorem with applications vol.29, pp.2, 2018, https://doi.org/10.1080/10652469.2017.1414214