Korean J. Math. Vol. 24 No. 3 (2016) pp.573-586
DOI: https://doi.org/10.11568/kjm.2016.24.3.573

Conditional integral transforms and convolutions for a general vector-valued conditioning functions

Article Sidebar

PDF
Published: Sep 30, 2016
Keywords:
conditional Wiener integral, conditional integral transform, conditional convolution, first variation
Mathematics Subject Classification:
28C20

Main Article Content

Bong Jin Kim
Department of Mathematics Daejin University Pocheon 11159, Korea
Byoung Soo Kim
School of Liberal Arts Seoul National University of Science and Technology Seoul 01811, Korea

Abstract

We study the conditional integral transforms and conditional convolutions of functionals defined on $K[0, T]$. We consider a general vector-valued conditioning functions $X_k (x) = \left(\gamma_1 (x), \ldots, \gamma_k (x)\right)$ where $\gamma_j (x)$ are Gaussian random variables on the Wiener space which need not depend upon the values of $x$ at only finitely many points in $(0, T]$. We then obtain several relationships and formulas for the conditioning functions that exist among conditional integral transform, conditional convolution and first variation of functionals in $E_\sigma$.


Article Details

References

[1] R.H.Cameron, The first variation of an indefinite Wiener integral, Proc. Amer. Math. Soc. 2 (1951), 914–924. Google Scholar

[2] R.H.Cameron and W.T.Martin, Fourier-Wiener transforms of analytic function- als, Duke Math. J. 12 (1945), 489–507. Google Scholar

[3] R.H.Cameron and D.A.Storvick, An L2 analytic Fourier-Feynman transform, Michigan Math. J. 23 (1976), 1–30. Google Scholar

[4] D.M.Chung and D.Skoug, Conditional analytic Feynman integrals and a related Schr ̈odinger integral equation, Siam. J. Math. Anal. 20 (1989), 950–965. Google Scholar

[5] B.A.Fuks, Theory of analytic functions of several complex variables, Amer. Math. Soc., Providence, Rhodo Island, 1963 Google Scholar

[6] B.J.Kim and B.S.Kim Generalized conditional integral transforms, conditional convolutions ans first variations, Korean J.Math. 20 (1) (2012), 1–18. Google Scholar

[7] B.J.Kim, B.S.Kim and D.Skoug, Integral transforms, convolution products and first variations, Internat. J. Math. Math. Sci. 2004 (11) (2004), 579–598. Google Scholar

[8] B.J.Kim, B.S.Kim and D.Skoug, Conditional integral transforms, conditional convolution products and first variations, PanAmerican Math. J. 14 (2004), 27– 47. Google Scholar

[9] Y.J.Lee, Integral transforms of analytic functions on abstract Wiener spaces, J. Funct. Anal. 47 (1982), 153–164. Google Scholar

[10] I.Y.Lee, H.S.Chung and S.J.Chang Integration formulas for the conditional transform involving the first variation, Bull.Iranian Math. Soc. 41 (2015), 771– 783. Google Scholar

[11] C.Park and D.Skoug, Conditional Fourier-Feynman transforms and conditional convolution products, J. Korean Math. Soc. 38 (2001), 61–76. Google Scholar

[12] C.Park and D.Skoug, A simple formula for conditional Wiener integrals with applications, Pacific J. Math. 135 (1988), 381–394. Google Scholar

[13] C.Park and D.Skoug, Conditional Wiener integrals II, Pacific J. Math. 167 (1995), 293–312. Google Scholar

[14] D.Skoug and D.Storvick, A survey of results involving transforms and convolu- tions in function space, Rocky Mountain J. Math. 34 (2004), 1147–1175. Google Scholar