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.


Introduction
It has long been known that Wiener measure and Wiener measurability behave badly under the change of scale transformation [3] and under translations [2]. However Cameron and Storvick [5] for a rather large class of functionals expressed the analytic Feynman integral as a limit of Wiener integrals. In doing so, they discovered a rather nice change of scale formula for Wiener integrals on classical Wiener space [6]. In [20,22,23], Yoo, Yoon and Skoug extended these results to Yeh-Wiener space and to an abstract Wiener space.
This paper continues the study of a change of scale formula for Wiener integrals on an abstract Wiener space previously given in" [22].
Motivated by the work of Kallianpur and Bromley [17], we establish a change of scale formula for Wiener integrals, for a larger class than the Fresnel class studied in [22], on an abstract Wiener space. Results in [5,6,20,22,23] will then be corollaries of our results.

Definitions and preliminaries
Let H be a real separable infinite dimensional Hilbert space with inner product (.,.) and norm 11 . 11. Let III . III be a measurable norm on H with respect to the Gaussian cylinder set measure u on H. Let B denote the completion of H with respect to III . Ill. Let l denote the natural injection from H to B. The adjoint operator t* of t is one-to-one and maps B* continuously onto a dense subset of H* . By identifying H with H* and 11* with £*B* , we have a triple where (.,.) denotes the natural dual pairing between Band B* . Bya well-known result of Gross [13], (70£-1 has a unique countably additive extension m to the Borel (7 -algebra B (B) of B . The triple (H,B,m) is called an abstract Wiener space. For more detailed, see [12,17,18,19]. DEFINITION In particular,

Change of scale formulas
We begin this section with a key lemma for Wiener integral on an abstract Wiener space (H, B, m).
In the following theorem, for F E FAt ,A2' we express the analytic Feynman integral of F over B x B as the limit of a sequence of Wiener integrals.
Next, using the bounded convergence theorem, equation ( Proof. This follows from the fact that for F E F( B) the sequential Feynman integral of F is equal to its analytic Feynman integral [18].
Our main. result, namely a cha.p.ge of scale formula for Wiener integrals on a product abstract Wiener space now follows from Theorem 3.2 above. The Banach algebra FAl ,A 2 is not closed with respect to pointwise or even uniform convergence [16,p2], and thus its uniform closure F~1,A2 with respect to uniform convergence s-a.e. is a larger space than F A1 ,A 2 . Next we show that equation (3.4) holds for F E F~1,A2' Next, using Theorem 3.2, the iterated limit theorem and the dominated convetgence theorem,we obtain: In addition, the equation (3.10) holds for F E F( B) u.
Proof. Apply Theorem 3.5 and 3.6 after making the following choices: Al = the identity, A 2 = 0 and PI = p. With these choices and Lemma 3.1, we can easily obtain our corollary.
Finally we end this section by showing that the class of functions for which Theorem 3.5 (Theorem 3.6) and Corollary 3.7 hold is more extensive than F~A and F(B)u respectively. To evaluate the integral on the right side of (3.10), we apply some technique in the proof of Lemma 3.1 so that  (BN, B (BN))' Then (HN,BN,mN)          The following corollaries show that the class of functionals for which the above corollaries hold is more extensive than 5;"(11).