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2010 | 8 | 5 | 908-927

Tytuł artykułu

Operator-valued Feynman integral via conditional Feynman integrals on a function space

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Abstrakty

EN
Let C 0r [0; t] denote the analogue of the r-dimensional Wiener space, define X t: C r[0; t] → ℝ2r by X t (x) = (x(0); x(t)). In this paper, we introduce a simple formula for the conditional expectations with the conditioning function X t. Using this formula, we evaluate the conditional analytic Feynman integral for the functional $$ \Gamma _t \left( x \right) = exp \left\{ {\int_0^t {\theta \left( {s,x\left( s \right)} \right)d\eta \left( s \right)} } \right\}\varphi \left( {x\left( t \right)} \right) x \in C^r \left[ {0,t} \right] $$, where η is a complex Borel measure on [0, t], and θ(s, ·) and φ are the Fourier-Stieltjes transforms of the complex Borel measures on ℝr. We then introduce an integral transform as an analytic operator-valued Feynman integral over C r [0, t], and evaluate the integral transform for the function Γt via the conditional analytic Feynman integral as a kernel.

Wydawca

Czasopismo

Rocznik

Tom

8

Numer

5

Strony

908-927

Opis fizyczny

Daty

wydano
2010-10-01
online
2010-09-28

Twórcy

autor
  • Kyonggi University

Bibliografia

  • [1] Ash R.B., Real Analysis and Probability, Probability and Mathematical Statistics, 11, Academic Press, New York-London, 1972
  • [2] Cameron R.H., The translation pathology of Wiener space, Duke Math. J., 1954, 21, 623–627 http://dx.doi.org/10.1215/S0012-7094-54-02165-1
  • [3] Cameron R.H., Storvick D.A., An operator valued function space integral and a related integral equation, J. Math. Mech., 1968, 18(6), 517–552
  • [4] Cameron R.H., Storvick D.A., An operator-valued function-space integral applied to integrals of functions of class L 1, Proc. Lond. Math. Soc., 1973, 27(2), 345–360 http://dx.doi.org/10.1112/plms/s3-27.2.345
  • [5] Cameron R.H., Storvick D.A., Some Banach algebras of analytic Feynman integrable functionals, In: Analytic Functions, Kozubnik 1979, Lecture Notes in Math., 798, Springer, Berlin-New York, 1980, 18–67 http://dx.doi.org/10.1007/BFb0097256
  • [6] Chang K.S., Cho D.H., Song T.S., Yoo I., A conditional analytic Feynman integral over Wiener paths in abstract Wiener space, Int. Math. J., 2002, 2(9), 855–870
  • [7] Chang K.S., Cho D.H., Yoo I., Evaluation formulas for a conditional Feynman integral over Wiener paths in abstract Wiener space, Czechoslovak Math. J., 2004, 54(129)(1), 161–180 http://dx.doi.org/10.1023/B:CMAJ.0000027256.06816.1a
  • [8] Cho D.H., Integral transform as operator-valued Feynman integrals via conditional Feynman integrals over Wiener paths in abstract Wiener space, Integral Transforms Spec. Funct., 2005, 16(2), 107–130 http://dx.doi.org/10.1080/10652460410001672988
  • [9] Cho D.H., A simple formula for an analogue of conditional Wiener integrals and its applications, Trans. Amer. Math. Soc., 2008, 360(7), 3795–3811 http://dx.doi.org/10.1090/S0002-9947-08-04380-8
  • [10] Cho D.H., Conditional Feynman integral and Schrödinger integral equation on a function space, Bull. Aust. Math. Soc., 2009, 79(1), 1–22 http://dx.doi.org/10.1017/S0004972708000920
  • [11] Cho D.H., Evaluation formulas for an analogue of conditional analytic Feynman integrals over a function space, preprint
  • [12] Chung D.M., Park C., Skoug D., Operator-valued Feynman integrals via conditional Feynman integrals, Pacific J. Math., 1990, 146(1), 21–42
  • [13] Im M.K., Ryu K.S., An analogue of Wiener measure and its applications, J. Korean Math. Soc., 2002, 39(5), 801–819 http://dx.doi.org/10.4134/JKMS.2002.39.5.801
  • [14] Johnson G.W., Lapidus M.L., Generalized Dyson Series, Generalized Feynman Diagrams, the Feynman Integral and Feynman’s Operational Calculus, Mem. Amer. Math. Soc., 351, AMS, Providence, 1986
  • [15] Johnson G.W., Skoug D.L., The Cameron-Storvick function space integral: the L 1 theory, J. Math. Anal. Appl., 1975, 50(3), 647–667 http://dx.doi.org/10.1016/0022-247X(75)90017-7
  • [16] Kuo H.H., Gaussian Measures in Banach Spaces, Lecture Notes in Math., 463, Springer, Berlin-New York, 1975
  • [17] Laha R.G., Rohatgi V.K., Probability Theory, Wiley Ser. Probab. Stat., Wiley & Sons, New York-Chichester-Brisbane, 1979
  • [18] Ryu K.S., The Wiener integral over paths in abstract Wiener space, J. Korean Math. Soc., 1992, 29(2), 317–331
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Bibliografia

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