• Title, Summary, Keyword: generalized Wiener space

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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.

A CHANGE OF SCALE FORMULA FOR GENERALIZED WIENER INTEGRALS II

  • Kim, Byoung Soo;Song, Teuk Seob;Yoo, Il
    • Journal of the Chungcheong Mathematical Society
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    • v.26 no.1
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    • pp.111-123
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    • 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
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    • v.24 no.3
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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.

GENERALIZED WHITE NOISE FUNCTIONALS ON CLASSICAL WIENER SPACE

  • Lee, Yuh-Jia
    • Journal of the Korean Mathematical Society
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    • v.35 no.3
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    • pp.613-635
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    • 1998
  • In this note we reformulate the white noise calculus on the classical Wiener space (C', C). It is shown that most of the examples and operators can be redefined on C without difficulties except the Hida derivative. To overcome the difficulty, we find that it is sufficient to replace C by L$_2$[0,1] and reformulate the white noise on the modified abstract Wiener space (C', L$_2$[0, 1]). The generalized white noise functionals are then defined and studied through their linear functional forms. For applications, we reprove the Ito formula and give the existence theorem of one-side stochastic integrals with anticipating integrands.

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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.

A FRESNEL TYPE CLASS ON FUNCTION SPACE

  • Chang, Seung-Jun;Choi, Jae-Gil;Lee, Sang-Deok
    • The Pure and Applied Mathematics
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    • v.16 no.1
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    • pp.107-119
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    • 2009
  • In this paper we define a Banach algebra on very general function space induced by a generalized Brownian motion process rather than on Wiener space, but the Banach algebra can be considered as a generalization of Fresnel class defined on Wiener space. We then show that several interesting functions in quantum mechanic are elements of the class.

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GENERALIZED FOURIER-WIENER FUNCTION SPACE TRANSFORMS

  • Chang, Seung-Jun;Chung, Hyun-Soo
    • Journal of the Korean Mathematical Society
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    • v.46 no.2
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    • pp.327-345
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    • 2009
  • In this paper, we define generalized Fourier-Hermite functionals on a function space $C_{a,b}[0,\;T]$ to obtain a complete orthonormal set in $L_2(C_{a,b}[0,\;T])$ where $C_{a,b}[0,\;T]$ is a very general function space. We then proceed to give a necessary and sufficient condition that a functional F in $L_2(C_{a,b}[0,\;T])$ has a generalized Fourier-Wiener function space transform ${\cal{F}}_{\sqrt{2},i}(F)$ also belonging to $L_2(C_{a,b}[0,\;T])$.

A GENERALIZED SEQUENTIAL OPERATOR-VALUED FUNCTION SPACE INTEGRAL

  • Chang, Kun-Soo;Kim, Byoung-Soo;Park, Cheong-Hee
    • Journal of the Korean Mathematical Society
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    • v.40 no.1
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    • pp.73-86
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    • 2003
  • In this paper, we define a generalized sequential operator-valued function space integral by using a generalized Wiener measure. It is an extention of the sequential operator-valued function space integral introduced by Cameron and Storvick. We prove the existence of this integral for functionals which involve some product Borel measures.

THE GENERALIZED ANALOGUE OF WIENER MEASURE SPACE AND ITS PROPERTIES

  • Ryu, Kun-Sik
    • Honam Mathematical Journal
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    • v.32 no.4
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    • pp.633-642
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    • 2010
  • In this note, we introduce the definition of the generalized analogue of Wiener measure on the space C[a, b] of all real-valued continuous functions on the closed interval [a, b], give several examples of it and investigate some important properties of it - the Fernique theorem and the existence theorem of scale-invariant measurable subsets on C[a, b].