A simple formula for an analogue of conditional Wiener integrals and its applications. II

Cho, Dong Hyun. "A simple formula for an analogue of conditional Wiener integrals and its applications. II." Czechoslovak Mathematical Journal 59.2 (2009): 431-452. <http://eudml.org/doc/37933>.

@article{Cho2009,
abstract = {Let $C[0,T]$ denote the space of real-valued continuous functions on the interval $[0,T]$ with an analogue $w_\varphi $ of Wiener measure and for a partition $ 0=t_0< t_1< \cdots < t_n <t_\{n+1\}= T$ of $[0, T]$, let $X_n\: C[0,T]\rightarrow \mathbb \{R\}^\{n+1\}$ and $X_\{n+1\} \: C [0, T]\rightarrow \mathbb \{R\}^\{n+2\}$ be given by $X_n(x) = ( x(t_0), x(t_1), \cdots , x(t_n))$ and $X_\{n+1\} (x) = ( x(t_0), x(t_1), \cdots , x(t_\{n+1\}))$, respectively. In this paper, using a simple formula for the conditional $w_\varphi $-integral of functions on $C[0, T]$ with the conditioning function $X_\{n+1\}$, we derive a simple formula for the conditional $w_\varphi $-integral of the functions with the conditioning function $X_n$. As applications of the formula with the function $X_n$, we evaluate the conditional $w_\varphi $-integral of the functions of the form $F_m(x) = \int _0^T (x(t))^m d t$ for $x\in C[0, T]$ and for any positive integer $m$. Moreover, with the conditioning $X_n$, we evaluate the conditional $w_\varphi $-integral of the functions in a Banach algebra $\mathcal \{S\}_\{w_\varphi \}$ which is an analogue of the Cameron and Storvick’s Banach algebra $\mathcal \{S\}$. Finally, we derive the conditional analytic Feynman $w_\varphi $-integrals of the functions in $\mathcal \{S\}_\{w_\varphi \}$.},
author = {Cho, Dong Hyun},
journal = {Czechoslovak Mathematical Journal},
keywords = {analogue of Wiener measure; Cameron-Martin translation theorem; conditional analytic Feynman $w_\varphi $-integral; conditional Wiener integral; Kac-Feynman formula; simple formula for conditional $w_\varphi $-integral; analogue of Wiener measure; Cameron-Martin translation theorem; conditional analytic Feynman -integral; conditional Wiener integral; Kac-Feynman formula; simple formula for conditional -integral},
language = {eng},
number = {2},
pages = {431-452},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A simple formula for an analogue of conditional Wiener integrals and its applications. II},
url = {http://eudml.org/doc/37933},
volume = {59},
year = {2009},
}

TY - JOUR
AU - Cho, Dong Hyun
TI - A simple formula for an analogue of conditional Wiener integrals and its applications. II
JO - Czechoslovak Mathematical Journal
PY - 2009
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 59
IS - 2
SP - 431
EP - 452
AB - Let $C[0,T]$ denote the space of real-valued continuous functions on the interval $[0,T]$ with an analogue $w_\varphi $ of Wiener measure and for a partition $ 0=t_0< t_1< \cdots < t_n <t_{n+1}= T$ of $[0, T]$, let $X_n\: C[0,T]\rightarrow \mathbb {R}^{n+1}$ and $X_{n+1} \: C [0, T]\rightarrow \mathbb {R}^{n+2}$ be given by $X_n(x) = ( x(t_0), x(t_1), \cdots , x(t_n))$ and $X_{n+1} (x) = ( x(t_0), x(t_1), \cdots , x(t_{n+1}))$, respectively. In this paper, using a simple formula for the conditional $w_\varphi $-integral of functions on $C[0, T]$ with the conditioning function $X_{n+1}$, we derive a simple formula for the conditional $w_\varphi $-integral of the functions with the conditioning function $X_n$. As applications of the formula with the function $X_n$, we evaluate the conditional $w_\varphi $-integral of the functions of the form $F_m(x) = \int _0^T (x(t))^m d t$ for $x\in C[0, T]$ and for any positive integer $m$. Moreover, with the conditioning $X_n$, we evaluate the conditional $w_\varphi $-integral of the functions in a Banach algebra $\mathcal {S}_{w_\varphi }$ which is an analogue of the Cameron and Storvick’s Banach algebra $\mathcal {S}$. Finally, we derive the conditional analytic Feynman $w_\varphi $-integrals of the functions in $\mathcal {S}_{w_\varphi }$.
LA - eng
KW - analogue of Wiener measure; Cameron-Martin translation theorem; conditional analytic Feynman $w_\varphi $-integral; conditional Wiener integral; Kac-Feynman formula; simple formula for conditional $w_\varphi $-integral; analogue of Wiener measure; Cameron-Martin translation theorem; conditional analytic Feynman -integral; conditional Wiener integral; Kac-Feynman formula; simple formula for conditional -integral
UR - http://eudml.org/doc/37933
ER -