• 충청수학회지
• /
• 제25권4호
• /
• pp.691-701
• /
• 2012

In this note, we define the It$\hat{o}$ integral with respect to analogue of Wiener process and investigate its various properties and examples.

• East Asian mathematical journal
• /
• 제31권1호
• /
• pp.135-144
• /
• 2015

In this paper we establish several formulas for multiple integral transform of functionals defined on abstract Wiener space. We then use the these results to establish several basic formulas involving multiple convolution products.

• Korean Journal of Mathematics
• /
• 제22권1호
• /
• pp.57-69
• /
• 2014

We obtain a formula for the conditional Wiener integral of the first variation of functionals and establish several integration by parts formulas of conditional Wiener integrals of functionals on a function space. We then apply these results to obtain various integration by parts formulas involving conditional integral transforms and conditional convolution products on the function space.

• 충청수학회지
• /
• 제19권1호
• /
• pp.61-68
• /
• 2006

In 2002, the author introduced the definition and its properties of an analogue of Wiener measure over paths. In this article, using these concepts, we will derive an operator-valued measure over paths and will investigate the properties for integral with respect to the measure. Specially, we will prove the Wiener integral formula for our integral and give some example of it.

• 대한수학회보
• /
• 제36권4호
• /
• pp.809-822
• /
• 1999

In this paper, we introduce conditional Yeh-Wiener in-tegrals for generalized conditioning functions including vector-valued functions. And also we establish various evaluation formulas of conditional Yeh-Wiener integrals for generalized conditioning functions.

• 대한수학회지
• /
• 제53권3호
• /
• pp.709-723
• /
• 2016

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 $Z_n:C[0,t]{\rightarrow}{\mathbb{R}}^n$ by $Zn(x)=(\int_{0}^{t_1}h(s)dx(s),{\cdots},\int_{0}^{t_n}h(s)dx(s))$, where 0 < $t_1$ < ${\cdots}$ < $t_n$ < t is a partition of [0, t] and $h{\in}L_2[0,t]$ with $h{\neq}0$ a.e. In this paper we will introduce a simple formula for a generalized conditional Wiener integral on C[0, t] with the conditioning function $Z_n$ and then evaluate the generalized analytic conditional Wiener and Feynman integrals of the cylinder function $F(x)=f(\int_{0}^{t}e(s)dx(s))$ for $x{\in}C[0,t]$, where $f{\in}L_p(\mathbb{R})(1{\leq}p{\leq}{\infty})$ and e is a unit element in $L_2[0,t]$. Finally we express the generalized analytic conditional Feynman integral of F as two kinds of limits of non-conditional generalized Wiener integrals of polygonal functions and of cylinder functions using a change of scale transformation for which a normal density is the kernel. The choice of a complete orthonormal subset of $L_2[0,t]$ used in the transformation is independent of e and the conditioning function $Z_n$ does not contain the present positions of the generalized Wiener paths.

• 대한수학회보
• /
• 제36권3호
• /
• pp.441-450
• /
• 1999

In this paper we first evaluate the conditional Wiener integral of certain functionals using a Park and Skoug's formula. and we also evaluate the conditional wiener integral E(F│$X_\alpha$) of functional F on C[0, T] given by $F(x)=exp\{{\int_0}^T s^kx(s)ds\}$ for a general conditioning function $X_\alpha$ on C[0,T].

• 충청수학회지
• /
• 제26권1호
• /
• pp.111-123
• /
• 2013

Cameron and Storvick discovered change of scale formulas for Wiener integrals on classical Wiener space. Yoo and Skoug extended this result to an abstract Wiener space. In this paper, we investigate a change of scale formula for generalized Wiener integrals of various functions using the generalized Fourier-Feynman transform.

• Korean Journal of Mathematics
• /
• 제27권2호
• /
• pp.445-463
• /
• 2019

Let C[0, T] denote the space of continuous real-valued functions on [0, T]. On the space C[0, T], we introduce a Banach algebra of series of functions which are generalized Fourier-Stieltjes transforms of measures of finite variation on the product of simplex and Euclidean space. We evaluate analytic Feynman integrals of the functions in the Banach algebra which play significant roles in the Feynman integration theory and quantum mechanics.

• Korean Journal of Mathematics
• /
• 제24권3호
• /
• pp.489-494
• /
• 2016

The purpose of this paper is to treat the generalized conditional Yeh-Wiener integral for the sample path-valued conditioning function. As a special case of our results, we obtain the results in [6].