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

Open Mathematics (2010)

• Volume: 8, Issue: 5, page 908-927
• ISSN: 2391-5455

top

## Abstract

top
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.

## How to cite

top

Dong Cho. "Operator-valued Feynman integral via conditional Feynman integrals on a function space." Open Mathematics 8.5 (2010): 908-927. <http://eudml.org/doc/269685>.

@article{DongCho2010,
abstract = {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 $\Gamma \_t \left( x \right) = exp \left\lbrace \{\int \_0^t \{\theta \left( \{s,x\left( s \right)\} \right)d\eta \left( s \right)\} \} \right\rbrace \varphi \left( \{x\left( t \right)\} \right) x \in C^r \left[ \{0,t\} \right]$ , 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.},
author = {Dong Cho},
journal = {Open Mathematics},
keywords = {Analogue of Wiener measure; Conditional analytic Feynman integral; Conditional analytic Wiener integral; Operator-valued Feynman integral; Simple formula for conditional expectation; Wiener space; analogue of Wiener measure; conditional analytic Feynman integral; conditional analytic Wiener integral; operator-valued Feynman integral; simple formula for conditional expectation},
language = {eng},
number = {5},
pages = {908-927},
title = {Operator-valued Feynman integral via conditional Feynman integrals on a function space},
url = {http://eudml.org/doc/269685},
volume = {8},
year = {2010},
}

TY - JOUR
AU - Dong Cho
TI - Operator-valued Feynman integral via conditional Feynman integrals on a function space
JO - Open Mathematics
PY - 2010
VL - 8
IS - 5
SP - 908
EP - 927
AB - 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 $\Gamma _t \left( x \right) = exp \left\lbrace {\int _0^t {\theta \left( {s,x\left( s \right)} \right)d\eta \left( s \right)} } \right\rbrace \varphi \left( {x\left( t \right)} \right) x \in C^r \left[ {0,t} \right]$ , 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.
LA - eng
KW - Analogue of Wiener measure; Conditional analytic Feynman integral; Conditional analytic Wiener integral; Operator-valued Feynman integral; Simple formula for conditional expectation; Wiener space; analogue of Wiener measure; conditional analytic Feynman integral; conditional analytic Wiener integral; operator-valued Feynman integral; simple formula for conditional expectation
UR - http://eudml.org/doc/269685
ER -

## References

top
1. [1] Ash R.B., Real Analysis and Probability, Probability and Mathematical Statistics, 11, Academic Press, New York-London, 1972
2. [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 Zbl0057.09601
3. [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 Zbl0186.20701
4. [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 Zbl0264.28005
5. [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. [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 Zbl1275.28015
7. [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 Zbl1047.28008
8. [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 Zbl1147.28301
9. [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 Zbl1151.28017
10. [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 Zbl1215.28012
11. [11] Cho D.H., Evaluation formulas for an analogue of conditional analytic Feynman integrals over a function space, preprint Zbl1233.28006
12. [12] Chung D.M., Park C., Skoug D., Operator-valued Feynman integrals via conditional Feynman integrals, Pacific J. Math., 1990, 146(1), 21–42 Zbl0732.28008
13. [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 Zbl1017.28007
14. [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 Zbl0638.28009
15. [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. [16] Kuo H.H., Gaussian Measures in Banach Spaces, Lecture Notes in Math., 463, Springer, Berlin-New York, 1975 Zbl0306.28010
17. [17] Laha R.G., Rohatgi V.K., Probability Theory, Wiley Ser. Probab. Stat., Wiley & Sons, New York-Chichester-Brisbane, 1979
18. [18] Ryu K.S., The Wiener integral over paths in abstract Wiener space, J. Korean Math. Soc., 1992, 29(2), 317–331 Zbl0768.28005
19. [19] Ryu K.S., Im M.K., A measure-valued analogue of Wiener measure and the measure-valued Feynman-Kac formula, Trans. Amer. Math. Soc., 2002, 354(12), 4921–4951 http://dx.doi.org/10.1090/S0002-9947-02-03077-5 Zbl1017.28008

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.