DOI QR코드

DOI QR Code

EVALUATION FORMULAS FOR AN ANALOGUE OF CONDITIONAL ANALYTIC FEYNMAN INTEGRALS OVER A FUNCTION SPACE

  • 투고 : 2009.11.10
  • 발행 : 2011.05.31

초록

Let $C^r$[0,t] be the function space of the vector-valued continuous paths x : [0,t] ${\rightarrow}$ $R^r$ and define $X_t$ : $C^r$[0,t] ${\rightarrow}$ $R^{(n+1)r}$ and $Y_t$ : $C^r$[0,t] ${\rightarrow}$ $R^{nr}$ by $X_t(x)$ = (x($t_0$), x($t_1$), ..., x($t_{n-1}$), x($t_n$)) and $Y_t$(x) = (x($t_0$), x($t_1$), ..., x($t_{n-1}$)), respectively, where 0 = $t_0$ < $t_1$ < ... < $t_n$ = t. In the present paper, with the conditioning functions $X_t$ and $Y_t$, we introduce two simple formulas for the conditional expectations over $C^r$[0,t], an analogue of the r-dimensional Wiener space. We establish evaluation formulas for the analogues of the analytic Wiener and Feynman integrals for the function $G(x)=\exp{{\int}_0^t{\theta}(s,x(s))d{\eta}(s)}{\psi}(x(t))$, where ${\theta}(s,{\cdot})$ and are the Fourier-Stieltjes transforms of the complex Borel measures on ${\mathbb{R}}^r$. Using the simple formulas, we evaluate the analogues of the conditional analytic Wiener and Feynman integrals of the functional G.

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참고문헌

  1. R. H. Cameron and D. A. Storvick, Some Banach algebras of analytic Feynman integrable functionals, Analytic functions, Kozubnik 1979 (Proc. Seventh Conf., Kozubnik, 1979), pp. 18-67, Lecture Notes in Math., 798, Springer, Berlin-New York, 1980. https://doi.org/10.1007/BFb0097256
  2. K. S. Chang, D. H. Cho, and I. Yoo, Evaluation formulas for a conditional Feynman integral over Wiener paths in abstract Wiener space, Czechoslovak Math. J. 54(129) (2004), no. 1, 161-180.
  3. D. H. Cho, Conditional analytic Feynman integral over product space of Wiener paths in abstract Wiener space, Rocky Mountain J. Math. 38 (2008), no. 1, 61-90. https://doi.org/10.1216/RMJ-2008-38-1-61
  4. D. H. Cho, A simple formula for an analogue of conditional Wiener integrals and its applications , Trans. Amer. Math. Soc. 360 (2008), no. 7, 3795-3811. https://doi.org/10.1090/S0002-9947-08-04380-8
  5. D. H. Cho, Conditional Feynman integral and Schrodinger integral equation on a function space, Bull. Aust. Math. Soc. 79 (2009), no. 1, 1-22. https://doi.org/10.1017/S0004972708000920
  6. D. H. Cho, A simple formula for an analogue of conditional Wiener integrals and its applications. II, Czechoslovak Math. J. 59(134) (2009), no. 2, 431-452.
  7. M. K. Im and K. S. Ryu, 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
  8. G. W. Johnson and M. L. Lapidus, Generalized Dyson series, generalized Feynman diagrams, the Feynman integral and Feynman's operational calculus, Mem. Amer. Math. Soc. 62 (1986), no. 351, vi+78 pp.
  9. G. W. Johnson and D. L. Skoug, Notes on the Feynman integral, III: the Schroedinger equation, Pacific J. Math. 105 (1983), no. 2, 321-358. https://doi.org/10.2140/pjm.1983.105.321
  10. G. Kallianpur and C. Bromley, Generalized Feynman integrals using analytic continuation in several complex variables, Stochastic analysis and applications, 217-267, Adv. Probab. Related Topics, 7, Dekker, New York, 1984.
  11. H. H. Kuo, Gaussian measures in Banach spaces, Lecture Notes in Mathematics 463, Springer-Verlag, Berlin-New York, 1975.
  12. C. Park and D. L. Skoug, A simple formula for conditional Wiener integrals with applications , Pacific J. Math. 135 (1988), no. 2, 381-394. https://doi.org/10.2140/pjm.1988.135.381
  13. 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
  14. J. Yeh, Inversion of conditional expectations, Pacific J. Math. 52 (1974), 631-640. https://doi.org/10.2140/pjm.1974.52.631
  15. J. Yeh, Inversion of conditional Wiener integrals, Pacific J. Math. 59 (1975), no. 2, 623-638. https://doi.org/10.2140/pjm.1975.59.623
  16. J. Yeh, Transformation of conditional Wiener integrals under translation and the Cameron-Martin translation theorem, Tohoku Math. J. (2) 30 (1978), no. 4, 505-515. https://doi.org/10.2748/tmj/1178229910