Notes on the Feynman integral. I

Abstract

1* Introduction* In a recent paper (2), Cameron and Storvick treat a Banach algebra S of functions on Wiener space which are a kind of stochastic Fourier transform of Borel measures on L2(a, &). (Precise definitions will be given in the next section.) For such functions they show that the analytic Feynman integral, defined by analytic continuation of the Wiener integral, exists, and they give a formula for this Feynman integral. The work in (2) is related to Albeverio and Hώegh-Krohn's beautiful theory (1) of infinite dimensional oscillatory integrals (Fresnel integrals) as well as to (5). Cameron and Storvick's work is highly promising and has some appealing features. For example, as we will show in a later note, the existence of the Feynman integral for certain qudratic potentials can be established without having to construct special spaces, quad- ratic forms, etc. to fit the particular problem of interest. The main purpose of this note is to show that a crucial part of (2) can be substantially simplified. Let R, C denote the real and complex numbers respectively. Let θ map (α, b) x R to C. Let C(a, b) denote Wiener space; that is, the space of R-valued continuous functions on (α, b) which vanish at α. Let m denote Wiener measure on C(a, &). Under certain hypotheses on θ, Cameron and Storvick show that the function