• Title, Summary, Keyword: change of scale formula

Search Result 35, Processing Time 0.06 seconds

RELATIONSHIP BETWEEN THE WIENER INTEGRAL AND THE ANALYTIC FEYNMAN INTEGRAL OF CYLINDER FUNCTION

  • Kim, Byoung Soo
    • Journal of the Chungcheong Mathematical Society
    • /
    • v.27 no.2
    • /
    • pp.249-260
    • /
    • 2014
  • Cameron and Storvick discovered a change of scale formula for Wiener integral of functionals in a Banach algebra $\mathcal{S}$ on classical Wiener space. We express the analytic Feynman integral of cylinder function as a limit of Wiener integrals. Moreover we obtain the same change of scale formula as Cameron and Storvick's result for Wiener integral of cylinder function. Our result cover a restricted version of the change of scale formula by Kim.

A CHANGE OF SCALE FORMULA FOR GENERALIZED WIENER INTEGRALS II

  • Kim, Byoung Soo;Song, Teuk Seob;Yoo, Il
    • Journal of the Chungcheong Mathematical Society
    • /
    • v.26 no.1
    • /
    • 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.

A CHANGE OF SCALE FORMULA FOR ENERALIZED WIENER INTEGRALS

  • Kim, Byoung Soo;Song, Teuk Seob;Yoo, Il
    • 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.

CHANGE OF SCALE FORMULAS FOR CONDITIONAL WIENER INTEGRALS AS INTEGRAL TRANSFORMS OVER WIENER PATHS IN ABSTRACT WIENER SPACE

  • Cho, Dong-Hyun
    • Communications of the Korean Mathematical Society
    • /
    • v.22 no.1
    • /
    • pp.91-109
    • /
    • 2007
  • In this paper, we derive a change of scale formula for conditional Wiener integrals, as integral transforms, of possibly unbounded functions over Wiener paths in abstract Wiener space. In fact, we derive the change of scale formula for the product of the functions in a Banach algebra which is equivalent to both the Fresnel class and the space of measures of bounded variation over a real separable Hilbert space, and the $L_p-type$cylinder functions over Wiener paths in abstract Wiener space. As an application of the result, we obtain a change of scale formula for the conditional analytic Fourier-Feynman transform of the product of the functions.

A CHANGE OF SCALE FORMULA FOR CONDITIONAL WIENER INTEGRALS ON CLASSICAL WIENER SPACE

  • Yoo, Il;Chang, Kun-Soo;Cho, Dong-Hyun;Kim, Byoung-Soo;Song, Teuk-Seob
    • 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.

A CHANGE OF SCALE FORMULA FOR WIENER INTEGRALS OF UNBOUNDED FUNCTIONS II

  • Yoo, Il;Song, Teuk-Seob;Kim, Byoung-Soo
    • 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.

A CHANGE OF SCALE FORMULA FOR WIENER INTEGRALS ON THE PRODUCT ABSTRACT WIENER SPACES

  • Kim, Young-Sik;Ahn, Jae-Moon;Chang, Kun-Soo;Il Yoo
    • 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.

  • PDF

A CONDITIONAL FOURIER-FEYNMAN TRANSFORM AND CONDITIONAL CONVOLUTION PRODUCT WITH CHANGE OF SCALES ON A FUNCTION SPACE I

  • Cho, Dong Hyun
    • Bulletin of the Korean Mathematical Society
    • /
    • v.54 no.2
    • /
    • pp.687-704
    • /
    • 2017
  • Using a simple formula for conditional expectations over an analogue of Wiener space, we calculate a generalized analytic conditional Fourier-Feynman transform and convolution product of generalized cylinder functions which play important roles in Feynman integration theories and quantum mechanics. We then investigate their relationships, that is, the conditional Fourier-Feynman transform of the convolution product can be expressed in terms of the product of the conditional FourierFeynman transforms of each function. Finally we establish change of scale formulas for the generalized analytic conditional Fourier-Feynman transform and the conditional convolution product. In this evaluation formulas and change of scale formulas we use multivariate normal distributions so that the orthonormalization process of projection vectors which are essential to establish the conditional expectations, can be removed in the existing conditional Fourier-Feynman transforms, conditional convolution products and change of scale formulas.

CHANGE OF SCALE FORMULAS FOR WIENER INTEGRAL OVER PATHS IN ABSTRACT WIENER SPACE

  • Kim, Byoung-Soo;Kim, Tae-Soo
    • Communications of the Korean Mathematical Society
    • /
    • v.21 no.1
    • /
    • pp.75-88
    • /
    • 2006
  • Wiener measure and Wiener measurability behave badly under the change of scale transformation. We express the analytic Feynman integral over $C_0(B)$ as a limit of Wiener integrals over $C_0(B)$ and establish change of scale formulas for Wiener integrals over $C_0(B)$ for some functionals.

CHANGE OF SCALE FORMULAS FOR A GENERALIZED CONDITIONAL WIENER INTEGRAL

  • Cho, Dong Hyun;Yoo, Il
    • Bulletin of the Korean Mathematical Society
    • /
    • v.53 no.5
    • /
    • pp.1531-1548
    • /
    • 2016
  • Let C[0, t] denote the space of real-valued continuous functions on [0, t] and define a random vector $Z_n:C[0,t]{\rightarrow}\mathbb{R}^n$ by $Z_n(x)=(\int_{0}^{t_1}h(s)dx(s),{\ldots},\int_{0}^{t_n}h(s)dx(s))$, 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. Using a simple formula for a conditional expectation on C[0, t] with $Z_n$, we evaluate a generalized analytic conditional Wiener integral of the function $G_r(x)=F(x){\Psi}(\int_{0}^{t}v_1(s)dx(s),{\ldots},\int_{0}^{t}v_r(s)dx(s))$ for F in a Banach algebra and for ${\Psi}=f+{\phi}$ which need not be bounded or continuous, where $f{\in}L_p(\mathbb{R}^r)(1{\leq}p{\leq}{\infty})$, {$v_1,{\ldots},v_r$} is an orthonormal subset of $L_2[0,t]$ and ${\phi}$ is the Fourier transform of a measure of bounded variation over $\mathbb{R}^r$. Finally we establish various change of scale transformations for the generalized analytic conditional Wiener integrals of $G_r$ with the conditioning function $Z_n$.