Abstract
Using simple formulas for generalized conditional Wiener integrals on a function space which is an analogue of Wiener space, we evaluate two generalized analytic conditional Wiener integrals of a generalized cylinder function which is useful in Feynman integration theories and quantum mechanics. We then establish various integral transforms over continuous paths with change of scales for the generalized analytic conditional Wiener integrals. In these evaluation formulas and integral transforms we use multivariate normal distributions so that the orthonormalization process of projection vectors which are needed to establish the conditional Wiener integrals can be removed in the existing change of scale transforms. Consequently the transforms in the present paper can be expressed in terms of the generalized cylinder function itself.
Highlights
Let C0[0, T] denote the classical Wiener space, the space of continuous real-valued functions x on [0, T] with x(0) = 0
Further change of scale formulas for conditional Wiener integrals was introduced by the author and his coauthors [8,9,10]
Change of scale formulas for conditional Wiener integrals was established on C0[0, T], on the infinite dimensional Wiener space, and on C[0, T], an analogue of Wiener space [11] which is the space of real-valued continuous paths on [0, T]
Summary
Some difficulties in studying the transforms for the conditional Wiener integrals of cylinder functions which play important roles in Feynman integration theories are that they cannot be expressed in terms of the original cylinder functions To avoid these difficulties, modified cylinder functions expressed by a polygonal function with projection vectors satisfying orthonormality were used to derive the change of scale transforms [8,9,10]. In this paper we use multivariate normal distributions so that the orthonormalization process of projection vectors which are needed to establish the conditional Wiener integrals can be removed in the existing change of scale transforms. We establish various change of scale transforms for the generalized analytic conditional Wiener integrals of FZ with Zn and Zn+1 In these evaluation formulas and scale transforms we use multivariate normal distributions so that Gram-Schmidt orthonormalization process of {P⊥hV1, . In contrast with the existing change of scale transforms in [8,9,10], the transforms in this paper are expressed in terms of the cylinder function FZ itself and generalize some results in those references
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