zbMATH — the first resource for mathematics

Generalized Fourier-Feynman transforms, convolution products, and first variations on function space. (English) Zbl 1202.60133
Let \(Y\) be a generalized Brownian motion process, i.e. a Gaussian process with an absolutely continuous mean function \(a(t)\), \(t\in [0,T]\), with \(a(0)=0\), \(a'\in L^2[0,T]\), and covariance function \(r(s,t)=\min\{b(s),b(t)\}\), \(s,t\in [0,T]\) where \(b\) is a strictly increasing \(C^1\)-function with \(b(0)=0\). The authors are interested in functionals of the form \[ F(x) = f(\langle\alpha_1,x \rangle, \ldots, \langle\alpha_n,x \rangle) \] where \(\langle\alpha,x \rangle = \int_0^T \alpha(t)\,dx(t)\) is the Paley-Wiener-Zygmund stochastic integral for \(x\in C_{a,b}[0,T]\). (Unfortunately, the authors fail to define this function space in the paper). The authors introduce the generalized analytic Fourier-Feynman transform, the generalized convolution product and the generalized first variation for such functionals \(F\). The main part of the paper is devoted to the study of various relationships if any two or any three distinct operations are combined.

60J65 Brownian motion
28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)
Full Text: DOI
[1] R.H. Cameron and D.A. Storvick, An \(L_2\) analytic Fourier-Feynman transform , Michigan Math. J. 23 (1976), 1-30. · Zbl 0382.42008
[2] ——–, Feynman integral of variations of functions, in Gaussian random fields , Ser. Prob. Statist. 1 (1991), 144-157. · Zbl 0820.46045
[3] K.S. Chang, B.S. Kim and I. Yoo, Fourier-Feynman transform, convolution and first variation of functionals on abstract Wiener space , Integral Transforms Special Functions 10 (2000), 179-200. · Zbl 0973.28011
[4] S.J. Chang, J.G. Choi, and D. Skoug, Integration by parts formulas involving generalized Fourier-Feynman transforms on function space , Trans. Amer. Math. Soc. 355 (2003), 2925-2948. JSTOR: · Zbl 1014.60077
[5] S.J. Chang and D. Skoug, The effect of drift on the Fourier-Feynman transform, the convolution product and the first variation , PanAmerican Math. J. 10 (2000), 25-38. · Zbl 0976.28008
[6] ——–, Generalized Fourier-Feynman transforms and a first variation on function space , Integral Transforms Special Functions 14 (2003), 375-393. · Zbl 1043.28014
[7] T. Huffman, C. Park and D. Skoug, Analytic Fourier-Feynman transforms and convolution , Trans. Amer. Math. Soc. 347 (1995), 661-673. JSTOR: · Zbl 0880.28011
[8] ——–, Generalized transforms and convolutions , Internat. J. Math. Math. Sci. 20 (1997), 19-32. · Zbl 0982.28011
[9] G.W. Johnson and D.L. Skoug, An \(L_p\) analytic Fourier-Feynman transform , Michigan Math. J. 26 (1979), 103-127. · Zbl 0409.28007
[10] ——–, Scale-invariant measurability in Wiener space , Pacific J. Math. 83 (1979), 157-176. · Zbl 0387.60070
[11] J.G. Kim, J.W. Ko, C. Park, and D. Skoug, Relationships among transforms, convolutions, and first variations , Internat. J. Math. Math. Sci. 22 (1999), 191-204. · Zbl 1030.28007
[12] C. Park, and D. Skoug, Integration by parts formulas involving analytic Feynman integrals , PanAmerican Math. J. 8 (1998), 1-11. · Zbl 0958.46042
[13] J. Yeh, Convolution in Fourier-Wiener transform , Pacific J. Math. 15 (1965), 731-738. · Zbl 0128.33702
[14] ——–, Stochastic processes and the Wiener integral , Marcel Dekker, Inc., New York, 1973. · Zbl 0277.60018
[15] I. Yoo, Convolution and the Fourier-Wiener transform on abstract Wiener space , Rocky Mountain J. Math. 25 (1995), 1577-1587. · Zbl 0855.28006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.