Relationships between generalized Wiener integrals and conditional analytic Feynman integrals over continuous paths

Kim, Byoung Soo, and Cho, Dong Hyun. "Relationships between generalized Wiener integrals and conditional analytic Feynman integrals over continuous paths." Czechoslovak Mathematical Journal 67.3 (2017): 609-628. <http://eudml.org/doc/294139>.

@article{Kim2017,
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 $Z_n\colon C[0,t]\rightarrow \mathbb \{R\}^\{n+1\}$ 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 $C[0,t]$.},
author = {Kim, Byoung Soo, Cho, Dong Hyun},
journal = {Czechoslovak Mathematical Journal},
keywords = {analogue of Wiener space; analytic conditional Feynman integral; change of scale formula; conditional Wiener integral; Wiener integral},
language = {eng},
number = {3},
pages = {609-628},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Relationships between generalized Wiener integrals and conditional analytic Feynman integrals over continuous paths},
url = {http://eudml.org/doc/294139},
volume = {67},
year = {2017},
}

TY - JOUR
AU - Kim, Byoung Soo
AU - Cho, Dong Hyun
TI - Relationships between generalized Wiener integrals and conditional analytic Feynman integrals over continuous paths
JO - Czechoslovak Mathematical Journal
PY - 2017
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 67
IS - 3
SP - 609
EP - 628
AB - 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 $Z_n\colon C[0,t]\rightarrow \mathbb {R}^{n+1}$ 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 $C[0,t]$.
LA - eng
KW - analogue of Wiener space; analytic conditional Feynman integral; change of scale formula; conditional Wiener integral; Wiener integral
UR - http://eudml.org/doc/294139
ER -