Browse > Article
http://dx.doi.org/10.4134/CKMS.c200297

A CAMERON-STORVICK THEOREM ON C2a,b[0, T ] WITH APPLICATIONS  

Choi, Jae Gil (School of General Education Dankook University)
Skoug, David (Department of Mathematics University of Nebraska-Lincoln)
Publication Information
Communications of the Korean Mathematical Society / v.36, no.4, 2021 , pp. 685-704 More about this Journal
Abstract
The purpose of this paper is to establish a very general Cameron-Storvick theorem involving the generalized analytic Feynman integral of functionals on the product function space C2a,b[0, T]. The function space Ca,b[0, T] can be induced by the generalized Brownian motion process associated with continuous functions a and b. To do this we first introduce the class ${\mathcal{F}}^{a,b}_{A_1,A_2}$ of functionals on C2a,b[0, T] which is a generalization of the Kallianpur and Bromley Fresnel class ${\mathcal{F}}_{A_1,A_2}$. We then proceed to establish a Cameron-Storvick theorem on the product function space C2a,b[0, T]. Finally we use our Cameron-Storvick theorem to obtain several meaningful results and examples.
Keywords
Generalized analytic Feynman integral; product function space generalized Brownian motion process; Kallianpur and Bromley Fresnel class; Cameron-Storvick theorem;
Citations & Related Records
Times Cited By KSCI : 1  (Citation Analysis)
연도 인용수 순위
1 C. Park, D. Skoug, and D. Storvick, Fourier-Feynman transforms and the first variation, Rend. Circ. Mat. Palermo (2) 47 (1998), no. 2, 277-292. https://doi.org/10.1007/BF02844368   DOI
2 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. http://projecteuclid.org/euclid.ijm/1256052816   DOI
3 G. Kallianpur and C. Bromley, Generalized Feynman integrals using analytic continuation in several complex variables, in Stochastic Analysis and Applications, 217-267, Adv. Probab. Related Topics, 7, Dekker, New York, 1984.
4 G. W. Johnson and D. L. Skoug, An Lp analytic Fourier-Feynman transform, Michigan Math. J. 26 (1979), no. 1, 103-127. http://projecteuclid.org/euclid.mmj/1029002166
5 R. H. Cameron and D. A. Storvick, Feynman integral of variations of functionals, in Gaussian random fields (Nagoya, 1990), 144-157, Ser. Probab. Statist., 1, World Sci. Publ., River Edge, NJ, 1991.
6 G. W. Johnson, The equivalence of two approaches to the Feynman integral, J. Math. Phys. 23 (1982), no. 11, 2090-2096. https://doi.org/10.1063/1.525250   DOI
7 S. J. Chang, J. G. Choi, and D. Skoug, Generalized Fourier-Feynman transforms, convolution products, and first variations on function space, Rocky Mountain J. Math. 40 (2010), no. 3, 761-788. https://doi.org/10.1216/RMJ-2010-40-3-761   DOI
8 R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, emended edition, Dover Publications, Inc., Mineola, NY, 2010.
9 J. Yeh, Stochastic Processes and the Wiener Integral, Marcel Dekker, Inc., New York, 1973.
10 H.-H. Kuo, Integration by parts for abstract Wiener measures, Duke Math. J. 41 (1974), 373-379. http://projecteuclid.org/euclid.dmj/1077310406   DOI
11 H.-H. Kuo and Y.-J. Lee, Integration by parts formula and the Stein lemma on abstract Wiener space, Commun. Stoch. Anal. 5 (2011), no. 2, 405-418. https://doi.org/10.31390/cosa.5.2.10   DOI
12 S. J. Chang and D. Skoug, Generalized Fourier-Feynman transforms and a first variation on function space, Integral Transforms Spec. Funct. 14 (2003), no. 5, 375-393. https://doi.org/10.1080/1065246031000074425   DOI
13 R. H. Cameron, The first variation of an indefinite Wiener integral, Proc. Amer. Math. Soc. 2 (1951), 914-924. https://doi.org/10.2307/2031708   DOI
14 S. J. Chang and J. G. Choi, Effect of drift of the generalized Brownian motion process: an example for the analytic Feynman integral, Arch. Math. (Basel) 106 (2016), no. 6, 591-600. https://doi.org/10.1007/s00013-016-0899-x   DOI
15 I. Pierce and D. Skoug, Integration formulas for functionals on the function space Ca,b[0, T] involving Paley-Wiener-Zygmund stochastic integrals, PanAmer. Math. J. 18 (2008), no. 4, 101-112.
16 W. Rudin, Real and Complex Analysis, third edition, McGraw-Hill Book Co., New York, 1987.
17 S. J. Chang, J. G. Choi, and A. Y. Ko, Multiple generalized analytic Fourier-Feynman transform via rotation of Gaussian paths on function space, Banach J. Math. Anal. 9 (2015), no. 4, 58-80. https://doi.org/10.15352/bjma/09-4-4   DOI
18 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   DOI
19 S. J. Chang, J. G. Choi, and D. Skoug, Evaluation formulas for conditional function space integrals. II, PanAmer. Math. J. 20 (2010), no. 3, 1-25.
20 M. D. Donsker, On function space integrals, in Analysis in Function Space, 17-30, M.I.T. Press, Cambridge, MA, 1964.
21 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   DOI
22 S. J. Chang, J. G. Choi, and D. Skoug, Translation theorems for the Fourier-Feynman transform on the product function space C2a,b[0, T], Banach J. Math. Anal. 13 (2019), no. 1, 192-216. https://doi.org/10.1215/17358787-2018-0022   DOI
23 K. S. Chang, T. S. Song, and I. Yoo, Analytic Fourier-Feynman transform and first variation on abstract Wiener space, J. Korean Math. Soc. 38 (2001), no. 2, 485-501.
24 J. G. Choi and S. J. Chang, Generalized Fourier-Feynman transform and sequential transforms on function space, J. Korean Math. Soc. 49 (2012), no. 5, 1065-1082. https://doi.org/10.4134/JKMS.2012.49.5.1065   DOI
25 J. G. Choi, D. Skoug, and S. J. Chang, Generalized analytic Fourier-Feynman transform of functionals in a Banach algebra $\mathcal{F}^{a,b}_{A_1,A_2}$, J. Funct. Spaces Appl. 2013 (2013), Art. ID 954098, 12 pp. https://doi.org/10.1155/2013/954098   DOI
26 D. L. Cohn, Measure Theory, second edition, Birkhauser Advanced Texts: Basler Lehrbucher., Birkhauser/Springer, New York, 2013. https://doi.org/10.1007/978-1-4614-6956-8   DOI
27 G. B. Folland, Real Analysis, second edition, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1999.
28 G. W. Johnson and M. L. Lapidus, The Feynman integral and Feynman's operational calculus, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000.