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Let C 0r [0; t] denote the analogue of the r-dimensional Wiener space, define X t: C r[0; t] → ℝ2r by X t (x) = (x(0); x(t)). In this paper, we introduce a simple formula for the conditional expectations with the conditioning function X t. Using this formula, we evaluate the conditional analytic Feynman integral for the functional $${\Gamma }_t\left(x\right)=exp\left\{{\int }_0^t\theta \left(s,x\left(s\right)\right)d\eta \left(s\right)\right\}\varphi \left(x\left(t\right)\right)x\in C^r\left[0,t\right]$$ , where η is a complex Borel measure on [0, t], and θ(s, ·) and φ are the Fourier-Stieltjes transforms of the complex Borel measures on ℝr. We then introduce an integral...

Huffman, Park and Skoug established several results involving Fourier-Feynman transform and convolution for functionals in a Banach algebra S on the classical Wiener space. Chang, Kim and Yoo extended these results to abstract Wiener space for a more generalized Fresnel class $${ℱ}_{{𝒜}_1,{𝒜}_2}$$ A1,A2 than the Fresnel class $$ℱ$$ (B)which corresponds to the Banach algebra S. In this paper we study Fourier-Feynman transform, convolution and first variation of unbounded functionals on abstract Wiener space having the form...

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 $X_{n}C[0,T]\to {ℝ}^{n+1}$ and $X_{n+1}C[0,T]\to {ℝ}^{n+2}$ be given by $X_n\left(x\right)=(x\left(t_0\right),x\left(t_1\right),\cdots ,x\left(t_n\right))$ and $X_{n+1}\left(x\right)=(x\left(t_0\right),x\left(t_1\right),\cdots ,x\left(t_{n+1}\right))$, 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...

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 $Z_n:C[0,t]\to {ℝ}^{n+1}$ by $$Z_n\left(x\right)=\left(x\left(0\right)+a\left(0\right),{\int }_0^{t_1}h\left(s\right)\mathrm{d}x\left(s\right)+x\left(0\right)+a\left(t_1\right),\cdots ,{\int }_0^{t_n}h\left(s\right)\mathrm{d}x\left(s\right)+x\left(0\right)+a\left(t_n\right)\right),$$ 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 $𝒮$. Finally, we express the generalized analytic conditional Feynman...

In this paper, we introduce a simple formula for conditional Wiener integrals over $C_0\left(𝔹\right)$, the space of abstract Wiener space valued continuous functions. Using this formula, we establish various formulas for a conditional Wiener integral and a conditional Feynman integral of functionals on $C_0\left(𝔹\right)$ in certain classes which correspond to the classes of functionals on the classical Wiener space introduced by Cameron and Storvick. We also evaluate the conditional Wiener integral and conditional Feynman integral...