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.

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

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

Classification (MSC 2010): 28C20, 60G05, 60G15, 60H05

DOI: 10.21136/CMJ.2017.0248-15

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