Analytic Fourier-Feynman Transforms and Convolution

Abstract

Abstract. In this paper we develop an Lp Fourier-Feynman theory for a class of functionals on Wiener space of the form F(x) = f(J0 axdx, ... , /0 a"dx). We then define a convolution product for functionals on Wiener space and show that the Fourier-Feynman transform of the convolution product is a product of Fourier-Feynman transforms.

Effect of drift of the generalized Brownian motion process: an example for the analytic Feynman integral

“…We then complete this function space to obtain the measure space [1][2][3]19,[22][23][24][27][28][29] if and only if a(t) ≡ 0…”

“…The functionals in S are defined as a stochastic Fourier transform of complex measures on L 2 [0, T ], and are bounded on C 0 [0, T ]. Other classes of the analytic Feynman integrable functionals on C 0 [0, T ] can be found in [2,19,[22][23][24][27][28][29]. But the 'analytic Feynman integral' cannot be interpreted as the integration in standard measure theory.…”

“…As mentioned above, the SBMP used in [1][2][3]19,[22][23][24][27][28][29] is stationary in time and is free of drift. However, the stochastic process used in this paper, as well as in [4][5][6][7][8][9][10][11][12][13][14][15][16]18,20,21,25,30], is non-stationary in time and is subject to a drift because In this note, we present an example of a bounded functional which is not generalized analytic Feynman integrable on the function space C a,b [0, T ].…”

L2-sequential transforms on function space

“…. , γ n ) can be regarded as a class of functionals having the form (6.5), and in [15], the authors extended the class A (2) n to the class A (2) n ≡ A (2) n (α 1 , . .…”

The Behavior of the First Variation Under the Fourier-Feynman Transform on Abstract Wiener Spaces

“…In 1976, R. H. Cameron and D. A. Storvick developed the L 2 analytic Fourier Feynman transform theory in [6] and in 1979, G. W. Johnson and D. L. Skoug developed the L p analytic Fourier Feynman transform theory in [11] for 1 ≤ p ≤ 2. In 1995-1997, T. Huffman and C. Park and D. A. Storvick established the behavior of the L p analytic Fourier-Feynman transform under the convolution in [8]- [10] and they showed that the Fourier-Feynman transform behaved nicely under the convolution and proving that the Fourier-Feynman transform of the convolution can be expressed as the product of Fourier-Feynman transforms of each functional. In 2001 and 2005, Y. S. Kim established the behavior of the Fourier-Feynman transforms under Wiener integrals in [13]- [15] and showing that the Fourier-Feynman transform can be nicely expressed as the limit of Wiener integrals and in [16], he showed that the Wiener integral over paths behave nicely under the change of scale on the abstract Wiener space.…”