Korean J. Math. Vol. 30 No. 2 (2022) pp.239-247
DOI: https://doi.org/10.11568/kjm.2022.30.2.239

Conditional Foruier-Feynman transform and convolution product for a vector valued conditioning function

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Bongjin Kim

Abstract

Let $C_0 [0,T]$ denote the Wiener space, the space of continuous functions $x(t)$ on $[0, T]$ such that $x(0)=0$. Define a random vector $Z_{{\vec e},k} : C_0 [0, T] \rightarrow {\mathbb R}^k$ by
$$ Z_{{\vec e},k} (x) =(\int_0^T e_1 (t) dx(t), \ldots,\int_0^T e_k (t) dx(t) )$$
where $e_j \in L_2 [0, T] $ with $e_j \ne 0$ a.e., $ j=1, \ldots , k$.
In this paper we study the conditional Fourier-Feynman transform and a conditional convolution product for a cylinder type functionals defined on $C_0 [0,T]$ with a general vector valued conditioning functions $Z_{{\vec e},k}$ above which need not depend upon the values of $x$ at only finitely many points in $(0, T]$ rather than a conditioning function $X(x) = (x(t_1 ) , \ldots, x(t_n ))$ where $0<t_1 < \ldots < t_n =T$. In particular we show that the conditional Fourier-Feynman transform of the conditional convolution product is the product of conditional Fourier-Feynman transforms.



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References

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

[2] K.S.Chang, D.H.Cho, B.S.Kim, T.S.Song and I.Yoo, Conditional Fourier-Feynman transform and convolution product over Wiener paths in abstract Wiener space, Integral Transforms and Special Functions, 14 (3) (2003), 217–235. Google Scholar

[3] S.J.Chang and J.G.Choi, Rotation of Gaussian paths on Wiener space with applications, Banach J. Math. Anal. 12 (3) (2018), 651–672. Google Scholar

[4] D.H.Cho, A generalized simple formula for evaluating Radon-Nikydym derivatives over paths, J. Korean Math. Soc. 58 (3) (2021), 609–631. Google Scholar

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

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

[7] .Huffman, C.Park and D.Skoug, Generalized transforms and convolutions, Internat J. Math. and Math Sci. 20 (1997), 19–32. Google Scholar

[8] B.J.Kim and B.S.Kim Conditional integral transforms and convolutions for a general vector-valued conditioning functions, Korean J. Math. 24 (2016), 573–586. Google Scholar

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

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

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