• Korean Journal of Mathematics
• /
• v.26 no.3
• /
• pp.387-403
• /
• 2018

Shifting, scaling and modulation proprerties for the convolution product of the Fourier-Feynman transform of functionals in a generalized Fresnel class ${\mathcal{F}}_{A1,A2}$ are given. These properties help us to obtain convolution product of new functionals from the convolution product of old functionals which we know their convolution product.

• Communications of the Korean Mathematical Society
• /
• v.12 no.3
• /
• pp.579-595
• /
• 1997

In this paper, we establish an $L_p$ analytic Fourier-Feynman transform theory for a class of cylinder functions on an abstract Wiener space. Also we define a convolution product for functions on an abstract Wiener space and then prove that the $L_p$ analytic Fourier-Feyman transform of the convolution product is a product of $L_p$ analytic Fourier-Feyman transforms.

• Communications of the Korean Mathematical Society
• /
• v.36 no.4
• /
• pp.685-704
• /
• 2021

The purpose of this paper is to establish a very general Cameron-Storvick theorem involving the generalized analytic Feynman integral of functionals on the product function space C²_a,b[0, T]. The function space C_a,b[0, T] can be induced by the generalized Brownian motion process associated with continuous functions a and b. To do this we first introduce the class ${\mathcal{F}}^{a,b}_{A_1,A_2}$ of functionals on C²_a,b[0, T] which is a generalization of the Kallianpur and Bromley Fresnel class ${\mathcal{F}}_{A_1,A_2}$. We then proceed to establish a Cameron-Storvick theorem on the product function space C²_a,b[0, T]. Finally we use our Cameron-Storvick theorem to obtain several meaningful results and examples.

• Journal of the Korean Mathematical Society
• /
• v.55 no.1
• /
• pp.147-160
• /
• 2018

In this article, we establish translation theorems for the analytic Fourier-Feynman transform of functionals in non-stationary Gaussian processes on Wiener space. We then proceed to show that these general translation theorems can be applied to two well-known classes of functionals; namely, the Banach algebra S introduced by Cameron and Storvick, and the space ${\mathcal{B}}^{(P)}_{\mathcal{A}}$ consisting of functionals of the form $F(x)=f({\langle}{\alpha}_1,x{\rangle},{\ldots},{\langle}{\alpha}_n,x{\rangle})$, where ${\langle}{\alpha},x{\rangle}$ denotes the Paley-Wiener-Zygmund stochastic integral ${\int_{0}^{T}}{\alpha}(t)dx(t)$.

• Journal of the Korean Mathematical Society
• /
• v.44 no.4
• /
• pp.1025-1050
• /
• 2007

Let $X_k(x)=({\int}^T_o{\alpha}_1(s)dx(s),...,{\int}^T_o{\alpha}_k(s)dx(s))\;and\;X_{\tau}(x)=(x(t_1),...,x(t_k))$ on the classical Wiener space, where ${{\alpha}_1,...,{\alpha}_k}$ is an orthonormal subset of $L_2$ [0, T] and ${\tau}:0 is a partition of [0, T]. In this paper, we establish a change of scale formula for conditional Wiener integrals $E[G_{\gamma}|X_k]$ of functions on classical Wiener space having the form $$G_{\gamma}(x)=F(x){\Psi}({\int}^T_ov_1(s)dx(s),...,{\int}^T_o\;v_{\gamma}(s)dx(s))$$, for $F{\in}S\;and\;{\Psi}={\psi}+{\phi}({\psi}{\in}L_p(\mathbb{R}^{\gamma}),\;{\phi}{\in}\hat{M}(\mathbb{R}^{\gamma}))$, which need not be bounded or continuous. Here S is a Banach algebra on classical Wiener space and $\hat{M}(\mathbb{R}^{\gamma})$ is the space of Fourier transforms of measures of bounded variation over $\mathbb{R}^{\gamma}$. As results of the formula, we derive a change of scale formula for the conditional Wiener integrals $E[G_{\gamma}|X_{\tau}]\;and\;E[F|X_{\tau}]$. Finally, we show that the analytic Feynman integral of F can be expressed as a limit of a change of scale transformation of the conditional Wiener integral of F using an inversion formula which changes the conditional Wiener integral of F to an ordinary Wiener integral of F, and then we obtain another type of change of scale formula for Wiener integrals of F.

• Kyungpook Mathematical Journal
• /
• v.52 no.4
• /
• pp.413-431
• /
• 2012

Let $C^r[0,t]$ be the function space of the vector-valued continuous paths $x:[0,t]{\rightarrow}\mathbb{R}^r$ and define $X_t:C^r[0,t]{\rightarrow}\mathbb{R}^{(n+1)r}$ and $Y_t:C^r[0,t]{\rightarrow}\mathbb{R}^{nr}$ by $X_t(x)=(x(t_0),\;x(t_1),\;{\cdots},\;x(t_{n-1}),\;x(t_n))$ and $Y_t(x)=(x(t_0),\;x(t_1),\;{\cdots},\;x(t_{n-1}))$, respectively, where $0=t_0$ < $t_1$ < ${\cdots}$ < $t_n=t$. In the present paper, using two simple formulas for the conditional expectations over $C^r[0,t]$ with the conditioning functions $X_t$ and $Y_t$, we establish evaluation formulas for the analogue of the conditional analytic Fourier-Feynman transform for the function of the form $${\exp}\{{\int_o}^t{\theta}(s,\;x(s))\;d{\eta}(s)\}{\psi}(x(t)),\;x{\in}C^r[0,t]$$ where ${\eta}$ is a complex Borel measure on [0, t] and both ${\theta}(s,{\cdot})$ and ${\psi}$ are the Fourier-Stieltjes transforms of the complex Borel measures on $\mathbb{R}^r$.

• Journal of the Chungcheong Mathematical Society
• /
• v.24 no.3
• /
• pp.517-528
• /
• 2011

Cameron and Storvick introduced change of scale formulas for Wiener integrals of bounded functions in the Banach algebra $\mathcal{S}$ of analytic Feynman integrable functions on classical Wiener space. Yoo and Skoug extended this result to an abstract Wiener space. Also Yoo, Song, Kim and Chang established a change of scale formula for Wiener integrals of functions on abstract Wiener space which need not be bounded or continuous. In this paper, we investigate a change of scale formula for generalized Wiener integrals of various functions on classical Wiener space.

• Journal of the Korean Mathematical Society
• /
• v.41 no.2
• /
• pp.319-344
• /
• 2004

In this paper, we define conditional integrals on abstract Wiener and Hilbert spaces and then obtain a formula for evaluating the integrals. We use this formula to establish the existence of conditional Feynman integrals for the classes $A^{q}$(B) and $A^{q}$(H) of functions on abstract Wiener and Hilbert spaces and then specialize this result to provide the fundamental solution to the Schrodinger equation with the forced harmonic oscillator.tor.

• Communications of the Korean Mathematical Society
• /
• v.21 no.1
• /
• pp.117-133
• /
• 2006

Cameron and Storvick discovered change of scale formulas for Wiener integrals of bounded functions in a Banach algebra S of analytic Feynman integrable functions on classical Wiener space. Yoo and Skoug extended these results to abstract Wiener space for a generalized Fresnel class $F_{A1,A2}$ containing the Fresnel class F(B) which corresponds to the Banach algebra S on classical Wiener space. In this paper, we present a change of scale formula for Wiener integrals of various functions on $B^2$ which need not be bounded or continuous.

• Journal of the Korean Mathematical Society
• /
• v.33 no.2
• /
• pp.269-282
• /
• 1996

It has long been known that Wiener measure and Wiener measurbility behave badly under the change of scale transformation [3] and under translation [2]. However, Cameron and Storvick [4] obtained the fact that the analytic Feynman integral was expressed as a limit of Wiener integrals for a rather larger class of functionals on a classical Wienrer space.