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
Classification MSC:  28C20, 60G05, 60G15, 60H05
DOI:  10.21136/CMJ.2017.0248-15


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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

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