A CHANGE OF SCALE FORMULA FOR WIENER INTEGRALS ON ABSTRACT WIENER SPACES

In this paper we obtain a change of scale formula for Wiener integrals on abstract Wiener spaces. This formula is shown to hold for many classes of functions of interest in Feynman integration theory and quantum mechanics

GENERALIZED TRANSFORMS AND CONVOLUTIONS

In this paper, using the concept of a generalized Feynman integral, we define a generalized Fourier-Feynman transform and a generalized convolution product. Then for two classes of functionals on Wiener space we obtain several results involving and relating these generalized transforms and convolutions. In particular we show that the generalized transform of the convolution product is a product of transforms. In addition we establish a Parseval’s identity for functionals in each of these classes

Analytic Fourier-Feynman Transforms And Convolution

In this paper we develop an Lp Fourier-Feynman theory for a class of functionals on Wiener space of the form F( x )= f (alJd x , . . . :J,l αndx ) . 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

FUNDAMENTAL THEOREM OF WIENER CALCULUS

In this paper we define and develop a theory of differentiation in Wiener space C[0,T]. We then proceed to establish a fundamental theorem of the integral calculus for C[0,T]. First of all, we show that the derivative of the indefinite Wiener integral exists and equals the integrand functional. Secondly, we show that certain functional defined on C[0,T] are equal to the indefinite integral of their Wiener derivative

Analytic Fourier-Feynman Transforms And Convolution

In this paper we develop an Lp Fourier-Feynman theory for a class of functionals on Wiener space of the form F( x )= f (alJd x , . . . :J,l αndx ) . 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

Translation Theorems for the Fourier-Feynman Transform on the Product Function Space C2 a,b, [0,T]

In this article, we establish the Cameron{Martin translation theo- rems for the analytic Fourier{Feynman transform of functionals on the product function space C2 a;b[0; T]. The function space Ca;b[0; T] is induced by the gener- alized Brownian motion process associated with continuous functions a(t) and b(t) on the time interval [0; T]. The process used here is nonstationary in time and is subject to a drift a(t). To study our translation theorem, we introduce a Fresnel-type class Fa;b A1;A2 of functionals on C2 a;b[0; T], which is a generaliza- tion of the Kallianpur and Bromley{Fresnel class FA1;A2 . We then proceed to establish the translation theorems for the functionals in Fa;b A1;A2

A change of scale formula for Wiener integrals on abstract Wiener spaces

In this paper we obtain a change of scale formula for Wiener integrals on abstract Wiener spaces. This formula is shown to hold for many classes of functions of interest in Feynman integration theory and quantum mechanics

RELATIONSHIPS AMONG TRANSFORMS, CONVOLUTIONS, AND FIRST VARIATIONS

In this paper, we establish several interesting relationships involving the Fourier-Feynman transform, the convolution product, and the first variation for functional F on Wiener space of the form F(x) = f((?α1,x),...,(αn,x)), where (αj,x) denotes the Paley-Wiener-Zygmund stochastic integral f0 αj(t)dx(t)