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

Title & Authors
GENERALIZED ANALYTIC FEYNMAN INTEGRAL VIA FUNCTION SPACE INTEGRAL OF BOUNDED CYLINDER FUNCTIONALS
Chang, Seung-Jun; Choi, Jae-Gil; Chung, Hyun-Soo;

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) = $\small{\hat{v}}$(($\small{g_1,x)^{\sim}}$,..., $\small{(g_n,x)^{\sim}}$) defined on a very general function space $\small{C_{a,b}}$[0,T]. We also present a change of scale formula for function space integrals of such cylinder functionals.
Keywords
generalized Brownian motion;generalized analytic Feynman integral;function space integral;cylinder functional;change of scale formula;
Language
English
Cited by
1.
GENERALIZED FOURIER-FEYNMAN TRANSFORM AND SEQUENTIAL TRANSFORMS ON FUNCTION SPACE,;;

대한수학회지, 2012. vol.49. 5, pp.1065-1082
1.
Change of path formula on the function space with applications, Indagationes Mathematicae, 2014, 25, 3, 596
2.
A Modified Analytic Function Space Feynman Integral and Its Applications, Journal of Function Spaces, 2014, 2014, 1
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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.