Czechoslovak Mathematical Journal, Vol. 67, No. 3, pp. 609-628, 2017
Relationships between generalized Wiener integrals and conditional analytic Feynman integrals over continuous paths
Byoung Soo Kim, Dong Hyun Cho
Received May 12, 2015. First published August 12, 2017.
Abstract: Let $C[0,t]$ denote a generalized Wiener space, the space of real-valued continuous functions on the interval $[0,t]$, and define a random vector $Z_n C[0,t]\to\mathbb R^{n+1}$ by Z_n(x)=\biggl(x(0)+a(0), \int_0^{t_1}h(s) {\rm d} x(s)+x(0)+a(t_1), \cdots,\int_0^{t_n}h(s) {\rm d} x(s)+x(0)+a(t_n)\biggr), where $a\in C[0,t]$, $h\in L_2[0,t]$, and $0<t_1 < \cdots< t_n\le t$ is a partition of $[0,t]$. Using simple formulas for generalized conditional Wiener integrals, given $Z_n$ we will evaluate the generalized analytic conditional Wiener and Feynman integrals of the functions $F$ in a Banach algebra which corresponds to Cameron-Storvick's Banach algebra $\mathcal S$. Finally, we express the generalized analytic conditional Feynman integral of $F$ as a limit of the non-conditional generalized Wiener integral of a polygonal function using a change of scale transformation for which a normal density is the kernel. This result extends the existing change of scale formulas on the classical Wiener space, abstract Wiener space and the analogue of the Wiener space $C[0,t]$.
Keywords: analogue of Wiener space; analytic conditional Feynman integral; change of scale formula; conditional Wiener integral; Wiener integral
References: [1] R. H. Cameron: The translation pathology of Wiener space. Duke Math. J. 21 (1954), 623-627. DOI 10.1215/S0012-7094-54-02165-1 | MR 0065033 | Zbl 0057.09601 [2] R. H. Cameron, W. T. Martin: The behavior of measure and measurability under change of scale in Wiener space. Bull. Am. Math. Soc. 53 (1947), 130-137. DOI 10.1090/S0002-9904-1947-08762-0 | MR 0019259 | Zbl 0032.41801 [3] R. H. Cameron, D. A. Storvick: Some Banach algebras of analytic Feynman integrable functionals. Analytic Functions Proc. Conf. Kozubnik 1979, Lect. Notes Math. 798, Springer, Berlin (1980), 18-67. DOI 10.1007/bfb0097256 | MR 0577446 | Zbl 0439.28007 [4] R. H. Cameron, D. A. Storvick: Change of scale formulas for Wiener integral. Functional Integration with Emphasis on the Feynman Integral Proc. Workshop Sherbrooke 1986, Suppl. Rend. Circ. Mat. Palermo, II. Ser. (1988), 105-115. MR 0950411 | Zbl 0653.28005 [5] K. S. Chang, D. H. Cho, I. Yoo: Evaluation formulas for a conditional Feynman integral over Wiener paths in abstract Wiener space. Czech. Math. J. 54 (2004), 161-180. DOI 10.1023/B:CMAJ.0000027256.06816.1a | MR 2040228 | Zbl 1047.28008 [6] D. H. Cho: Change of scale formulas for conditional Wiener integrals as integral transforms over Wiener paths in abstract Wiener space. Commun. Korean Math. Soc. 22 (2007), 91-109. DOI 10.4134/CKMS.2007.22.1.091 | MR 2286898 | Zbl 1168.28311 [7] D. H. Cho: A simple formula for a generalized conditional Wiener integral and its applications. Int. J. Math. Anal., Ruse 7 (2013), 1419-1431. DOI 10.12988/ijma.2013.3363 | MR 3066550 | Zbl 1285.28018 [8] D. H. Cho: Analogues of conditional Wiener integrals with drift and initial distribution on a function space. Abstr. Appl. Anal. (2014), Article ID 916423, 12 pages. DOI 10.1155/2014/916423 | MR 3226236 [9] D. H. Cho: Scale transformations for present position-dependent conditional expectations over continuous paths. Ann. Funct. Anal. AFA 7 (2016), 358-370. DOI 10.1215/20088752-3544830 | MR 3484389 | Zbl 1346.46038 [10] D. H. Cho: Scale transformations for present position-independent conditional expectations. J. Korean Math. Soc. 53 (2016), 709-723. DOI 10.4134/JKMS.j150285 | MR 3498289 | Zbl 1339.28019 [11] D. H. Cho, B. J. Kim, I. Yoo: Analogues of conditional Wiener integrals and their change of scale transformations on a function space. J. Math. Anal. Appl. 359 (2009), 421-438. DOI 10.1016/j.jmaa.2009.05.023 | MR 2546758 | Zbl 1175.28010 [12] D. H. Cho, I. Yoo: Change of scale formulas for a generalized conditional Wiener integral. Bull. Korean Math. Soc. 53 (2016), 1531-1548. DOI 10.4134/BKMS.b150795 | MR 3553416 | Zbl 1350.28015 [13] M. K. Im, K. S. Ryu: An analogue of Wiener measure and its applications. J. Korean Math. Soc. 39 (2002), 801-819. DOI 10.4134/JKMS.2002.39.5.801 | MR 1920906 | Zbl 1017.28007 [14] B. S. Kim: Relationship between the Wiener integral and the analytic Feynman integral of cylinder function. J. Chungcheong Math. Soc. 27 (2014), 249-260. DOI 10.14403/jcms.2014.27.2.249 [15] H.-H. Kuo: Gaussian Measures in Banach Spaces. Lecture Notes in Mathematics 463, Springer, Berlin (1975). DOI 10.1007/BFb0082007 | MR 0461643 | Zbl 0306.28010 [16] I. D. Pierce: On a Family of Generalized Wiener Spaces and Applications. Ph.D. Thesis, The University of Nebraska, Lincoln (2011). MR 2890101 [17] K. S. Ryu, M. K. Im: A measure-valued analogue of Wiener measure and the measure-valued Feynman-Kac formula. Trans. Am. Math. Soc. 354 (2002), 4921-4951. DOI 10.1090/S0002-9947-02-03077-5 | MR 1926843 | Zbl 1017.28008 [18] I. Yoo, K. S. Chang, D. H. Cho, B. S. Kim, T. S. Song: A change of scale formula for conditional Wiener integrals on classical Wiener space. J. Korean Math. Soc. 44 (2007), 1025-1050. DOI 10.4134/JKMS.2007.44.4.1025 | MR 2334543 | Zbl 1129.28014 [19] I. Yoo, D. Skoug: A change of scale formula for Wiener integrals on abstract Wiener spaces. Int. J. Math. Math. Sci. 17 (1994), 239-247. DOI 10.1155/S0161171294000359 | MR 1261069 | Zbl 0802.28008 [20] I. Yoo, D. Skoug: A change of scale formula for Wiener integrals on abstract Wiener spaces II. J. Korean Math. Soc. 31 (1994), 115-129. MR 1269456 | Zbl 0802.28009 [21] I. Yoo, T. S. Song, B. S. Kim, K. S. Chang: A change of scale formula for Wiener integrals of unbounded functions. Rocky Mt. J. Math. 34 (2004), 371-389. DOI 10.1216/rmjm/1181069911 | MR 2061137 | Zbl 1048.28010
Affiliations: Byoung Soo Kim, School of Liberal Arts, Seoul National University of Science and Technology, 232 Gongneung, Nowon, Seoul 01811, Republic of Korea, e-mail: mathkbs@seoultech.ac.kr; Dong Hyun Cho, Department of Mathematics, Kyonggi University, 154-42 Gwanggyosan, Iui, Yeongtong, Suwon 16227, Gyeonggi, Republic of Korea, e-mail: j94385@kyonggi.ac.kr
