• 대한수학회보
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• 제59권5호
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• pp.1093-1103
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• 2022

In this paper, we establish an evaluation formula to calculate the Wiener integral of polynomials in terms of natural dual pairings on abstract Wiener spaces (H, B, 𝜈). To do this we first derive a translation theorem for the Wiener integral of functionals associated with operators in 𝓛(B), the Banach space of bounded linear operators from B to itself. We then apply the translation theorem to establish an integration by parts formula for the Wiener integral of functionals combined with operators in 𝓛(B). We finally apply this parts formula to evaluate the Wiener integral of certain polynomials in terms of natural dual pairings.

• 대한수학회보
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• 제59권5호
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• pp.1131-1144
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• 2022

In this paper, we establish several basic formulas among the double-integral transforms, the double-convolution products, and the inverse double-integral transforms of cylinder functionals on abstract Wiener space. We then discuss possible relationships involving the double-integral transform.

• 대한수학회지
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• 제44권4호
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• pp.1025-1050
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• 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.

• 한국수학사학회지
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• 제13권2호
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• pp.65-72
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• 2000

In this paper we treat the Yeh-Wiener integral and the conditional Yeh-Wiener integral for vector-valued conditioning function which are examples of the function space integrals. Finally, we state the modified conditional Yeh-Wiener integral for vector-valued conditioning function.

• 한국수학사학회지
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• 제12권2호
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• pp.41-46
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• 1999

In this paper we first introduce the Wiener integral which is one of the function space integrals. And then we treat the conditional Wiener integral and explain the simple formula for the conditional Wiener integral with an example.

• 대한수학회보
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• 제47권1호
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• pp.131-149
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• 2010

In 1992, the author introduced the definition and the properties of Wiener measure over paths in Wiener space and this measure was investigated extensively by some mathematicians. In 2002, the author and Dr. Im presented an article for analogue of Wiener measure and its applications which is the generalized theory of Wiener measure theory. In this note, we will derive the analogue of Wiener measure over paths in Wiener space and establish two integration formulae, one is similar to the Wiener integration formula and another is similar to simple formula for conditional Wiener integral. Furthermore, we will give some examples for our formulae.

• 대한수학회보
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• 제36권3호
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• pp.419-439
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• 1999

In this paper, we first develop a general formula for evaluating conditional abstract Wiener integrals of cylinder functions. we next use our formula to evaluate the conditional abstract wiener integral of various cylinder functions and then specialize our results to conditional Yeh-Wiener integrals to show that we can obtain the corresponding results by Park and Skoug. We finally obtain a Cameron-Martin translation theorem for conditional abstract Wiener integrals.

• Kyungpook Mathematical Journal
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• 제49권1호
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• pp.155-166
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• 2009

We give a necessary and sufficient condition that a square integrable functional F(x) on Yeh-Wiener space has an integral transform $\hat{F}_{{\alpha},{\beta}}F(x)$ which is also square integrable. This extends the result by Kim and Skoug for functional F(x) in $L_2(C_0[0,T])$.

• 충청수학회지
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• 제24권3호
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• pp.517-528
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• 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.

• 대한수학회보
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• 제53권3호
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• pp.903-924
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• 2016

In the present paper, using a simple formula for the conditional expectations given a generalized conditioning function over an analogue of vector-valued Wiener space, we prove that the analytic operator-valued Feynman integrals of certain classes of functions over the space can be expressed by the conditional analytic Feynman integrals of the functions. We then provide the conditional analytic Feynman integrals of several functions which are the kernels of the analytic operator-valued Feynman integrals.