• Title, Summary, Keyword: Wiener measure

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INTEGRATION WITH RESPECT TO ANALOGUE OF WIENER MEASURE OVER PATHS IN WIENER SPACE AND ITS APPLICATIONS

  • Ryu, Kun-Sik
    • Bulletin of the Korean Mathematical Society
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    • v.47 no.1
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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.

THE SIMPLE FORMULA OF CONDITIONAL EXPECTATION ON ANALOGUE OF WIENER MEASURE

  • Ryu, Kun-Sik
    • Honam Mathematical Journal
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    • v.30 no.4
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    • pp.723-732
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    • 2008
  • In this note, we establish the uniqueness theorem of conditional expectation on analogue of Wiener measure space for given distributions and prove the simple formula of conditional expectation on analogue of Wiener measure which is essentially similar to Park and Skoug's formula on the concrete Wiener measure.

THE ROTATION THEOREM ON ANALOGUE OF WIENER SPACE

  • Ryu, Kun-Sik;Shim, Shung-Hoon
    • Honam Mathematical Journal
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    • v.29 no.4
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    • pp.577-588
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    • 2007
  • Bearman's rotation theorem is not only very important in pure mathematics but also plays the key role for various research areas, related to Wiener measure. In 2002, the author and professor Im introduced the concept of analogue of Wiener measure, a kind of generalization of Wiener measure and they presented the several papers associated with it. In this article, we prove a formula on analogue of Wiener measure, similar to the formula in Bearman's rotation theorem.

THE DOBRAKOV INTEGRAL OVER PATHS

  • Ryu, Kun Sik
    • Journal of the Chungcheong Mathematical Society
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    • v.19 no.1
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    • pp.61-68
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    • 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.

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THE TRANSFORMATION THEOREM ON ANALOGUE OF WIENER SPACE

  • Im, Man-Kyu
    • The Pure and Applied Mathematics
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    • v.14 no.4
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    • pp.317-333
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    • 2007
  • In 2002, the author and professor Ryu introduced the concept of analogue of Wiener measure. In this paper, we prove the existence theorem of Fourier-Feynman transform on analogue of Wiener measure in $L_2-norm$ sense.

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THE ARCSINE LAW IN THE GENERALIZED ANALOGUE OF WIENER SPACE

  • Ryu, Kun Sik
    • Journal of the Chungcheong Mathematical Society
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    • v.30 no.1
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    • pp.67-76
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    • 2017
  • In this note, we prove the theorems in the generalized analogue of Wiener space corresponding to the second and the third arcsine laws in either concrete or analogue of Wiener space [1, 2, 7] and we show that our results are exactly same to either the concrete or the analogue of Wiener case when the initial condition gives either the Dirac measure at the origin or the probability Borel measure.

EVALUATION E(exp(∫0th(s)dx(s)) ON ANALOGUE OF WIENER MEASURE SPACE

  • Park, Yeon-Hee
    • Honam Mathematical Journal
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    • v.32 no.3
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    • pp.441-451
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    • 2010
  • In this paper we evaluate the analogue of Wiener integral ${\int\limits}_{C[0,t]}x(t_1){\cdots}x(t_n)d\omega_\rho(x)$ where 0 = $t_0$ < $t_1$ $\cdots$ < $t_n$ $\leq$ t and the Paley-Wiener-Zygmund integral ${\int\limits}_{C[0,t]}$ exp $({\int\limits}_0^t h(s)\tilde{d}x(s))d\omega_\rho(x)$ is the analogue of Wiener measure space.

PROBABILITIES OF ANALOGUE OF WIENER PATHS CROSSING CONTINUOUSLY DIFFERENTIABLE CURVES

  • Ryu, Kun Sik
    • Journal of the Chungcheong Mathematical Society
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    • v.22 no.3
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    • pp.579-586
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    • 2009
  • Let $\varphi$ be a complete probability measure on $\mathbb{R}$, let $m_{\varphi}$ be the analogue of Wiener measure over paths on [0, T] and let f(t) be continuously differentiable on [0, T]. In this note, we give the analogue of Wiener measure $m_{\varphi}$ of {x in C[0, T]$\mid$x(0) < f(0) and $x(s_0){\geq}f(s_{0})$ for some $s_{0}$ in [0, T]} by use of integral equation techniques. This result is a generalization of Park and Paranjape's 1974 result[1].

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AN INTEGRATION FORMULA FOR ANALOGUE OF WIENER MEASURE AND ITS APPLICATIONS

  • Ryu, Kun Sik
    • Journal of the Chungcheong Mathematical Society
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    • v.23 no.4
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    • pp.711-720
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    • 2010
  • In this note, we will establish the integration formulae for functionals such as $F(x)=\prod_{j=1}^{n}\;x(s_j)^2$ and $G(x)=\exp\{{\lambda}{\int}_{0}^{t}\;x(s)^2dm_L(s)\}$ in the analogue of Wiener measure space and using our formulae, we will derive some formulae for series.