A Multiple Generalized Fourier–feynman Transform via a Rotation on Wiener Space

Abstract

In this paper we use a rotation property of Wiener measure to define a very general multiple Fourier-Feynman transform on Wiener space. We then proceed to establish its many algebraic properties as well as to establish several relationships between this generalized multiple transform and the corresponding generalized convolution product.

The behavior of conditional Wiener integrals on product Wiener space

“…In 8, the authors introduced a multiple analytic generalized Fourier‐Feynman transform on Wiener space and studied several algebraic properties for a class of the transforms. In this section we define a multiple conditional analytic Fourier‐Feynman transform (MCFFT) and a multiple conditional analytic convolution product (MCCP) of functionals on C 0 [0, T ].…”

“…Bearman's theorem was further developed by Cameron and Storvick 3 and by Johnson and Skoug 10 in their studies of Wiener integral equations. Recently in 8, using results in 7, Chang, Choi, and Skoug obtained results involving a very general multiple Fourier‐Feynman transform on Wiener space.…”

Parts formulas involving the Fourier–Feynman transform associated with Gaussian paths on Wiener space

“…where Z h (x, ·) is a Gaussian path given by the stochastic integral For a precise definition of this stochastic integral, see Section 2 below. Also the concept of the generalized analytic Feynman integral and the generalized analytic FFT (henceforth GFFT) were more developed based on the generalized Wiener integral (1.1), see [12,16,17,18,19,20,22,28]. If we choose h ≡ 1 in (1.2), as a constant function, the generalized Wiener integral (1.1) reduces an ordinary Wiener integral, i.e.,…”

L2-sequential transforms on function space

“…Then (C 0 [0, T ], M, m w ) is a complete measure space. The concepts of the generalized Wiener integral (the Wiener integral with respect to Gaussian paths) and the generalized analytic Feynman integral on C 0 [0, T ] were introduced by Chung, Park and Skoug [14], and further developed in [5,13,17]. In [5,13,14,17], the generalized Wiener integral is defined by the Wiener integral…”

“…The Wiener process W on C 0 [0, T ] × [0, T ] given by W (x, t) = x(t) is free of drift and is stationary in time while the Gaussian process Z h used in [5,13,14,17] is free of drift and is non-stationary in time. The stochastic process X on C a,b [0, T ] × [0, T ] defined by X(x, t) = x(t) is subject to a drift a(t) and is non-stationary in time.…”