• Title, Summary, Keyword: Yeh-Wiener space

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BOUNDARY-VALUED CONDITIONAL YEH-WIENER INTEGRALS AND A KAC-FEYNMAN WIENER INTEGRAL EQUATION

  • Park, Chull;David Skoug
    • Journal of the Korean Mathematical Society
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    • v.33 no.4
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    • pp.763-775
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    • 1996
  • For $Q = [0,S] \times [0,T]$ let C(Q) denote Yeh-Wiener space, i.e., the space of all real-valued continuous functions x(s,t) on Q such that x(0,t) = x(s,0) = 0 for every (s,t) in Q. Yeh [10] defined a Gaussian measure $m_y$ on C(Q) (later modified in [13]) such that as a stochastic process ${x(s,t), (s,t) \epsilon Q}$ has mean $E[x(s,t)] = \smallint_{C(Q)} x(s,t)m_y(dx) = 0$ and covariance $E[x(s,t)x(u,\upsilon)] = min{s,u} min{t,\upsilon}$. Let $C_\omega \equiv C[0,T]$ denote the standard Wiener space on [0,T] with Wiener measure $m_\omega$. Yeh [12] introduced the concept of the conditional Wiener integral of F given X, E(F$\mid$X), and for case X(x) = x(T) obtained some very useful results including a Kac-Feynman integral equation.

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A NOTE ON THE MODIFIED CONDITIONAL YEH-WIENER INTEGRAL

  • Chang, Joo-Sup;Ahn, Joong-Hyun
    • Communications of the Korean Mathematical Society
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    • v.16 no.4
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    • pp.627-635
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    • 2001
  • In this paper, we first introduce the modified Yeh-Wiener integral and then consider the modified conditional Yeh-Wiener integral. Here we use the space of continuous functions on a different region which was discussed before. We also evaluate some modified conditional Yeh-Wiener integral with examples using the simple formula for the modified conditional Yeh-Wiener integral.

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FOURIER-YEH-FEYNMAN TRANSFORM AND CONVOLUTION ON YEH-WIENER SPACE

  • Kim, Byoung Soo;Yang, Young Kyun
    • Korean Journal of Mathematics
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    • v.16 no.3
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    • pp.335-348
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    • 2008
  • We define Fourier-Yeh-Feynman transform and convolution product on the Yeh-Wiener space, and establish the existence of Fourier-Yeh-Feynman transform and convolution product for functionals in a Banach algebra $\mathcal{S}(Q)$. Also we obtain Parseval's relation for those functionals.

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CONDITIONAL ABSTRACT WIENER INTEGRALS OF CYLINDER FUNCTIONS

  • Chang, Seung-Jun;Chung, Dong-Myung
    • Bulletin of the Korean Mathematical Society
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    • v.36 no.3
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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.

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함수 공간 적분에 관한 소고(II)

  • 장주섭
    • Journal for History of Mathematics
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    • v.13 no.2
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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.

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EVALUATION OF SOME CONDITIONAL WIENER INTEGRALS

  • Chang, Kun-Soo;Chang, Joo-Sup
    • Bulletin of the Korean Mathematical Society
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    • v.21 no.2
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    • pp.99-106
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    • 1984
  • J. Yeh has recently introduced the concept of conditional Wiener integrals which are meant specifically the conditional expectation E$^{w}$ (Z vertical bar X) of a real or complex valued Wiener integrable functional Z conditioned by the Wiener measurable functional X on the Wiener measure space (A precise definition of the conditional Wiener integral and a brief discussion of the Wiener measure space are given in Section 2). In [3] and [4] he derived some inversion formulae for conditional Wiener integrals and evaluated some conditional Wiener integrals E$^{w}$ (Z vertical bar X) conditioned by X(x)=x(t) for a fixed t>0 and x in Wiener space. Thus E$^{w}$ (Z vertical bar X) is a real or complex valued function on R$^{1}$. In this paper we shall be concerned with the random vector X given by X(x) = (x(s$_{1}$),..,x(s$_{n}$ )) for every x in Wiener space where 0=s$_{0}$ $_{1}$<..$_{n}$ =t. In Section 3 we will evaluate some conditional Wiener integrals E$^{w}$ (Z vertical bar X) which are real or complex valued functions on the n-dimensional Euclidean space R$^{n}$ . Thus we extend Yeh's results [4] for the random variable X given by X(x)=x(t) to the random vector X given by X(x)=(x(s$_{1}$).., x(s$_{n}$ )).

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REFLECTION PRINCIPLES FOR GENERAL WIENER FUNCTION SPACES

  • Pierce, Ian;Skoug, David
    • Journal of the Korean Mathematical Society
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    • v.50 no.3
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    • pp.607-625
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    • 2013
  • It is well-known that the ordinary single-parameter Wiener space exhibits a reflection principle. In this paper we establish a reflection principle for a generalized one-parameter Wiener space and apply it to the integration of a class of functionals on this space. We also discuss several notions of a reflection principle for the two-parameter Wiener space, and explore whether these actually hold.