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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Homogeneous chaos, $ p$-forms, scaling and the Feynman integral

Authors: G. W. Johnson and G. Kallianpur
Journal: Trans. Amer. Math. Soc. 340 (1993), 503-548
MSC: Primary 60G15; Secondary 28C20, 46G12, 46N30, 60H05, 60H30, 60J65, 81S40
MathSciNet review: 1124168
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Abstract: In a largely heuristic but fascinating recent paper, Hu and Meyer have given a "formula" for the Feynman integral of a random variable $ f$ on Wiener space in terms of the expansion of $ f$ in Wiener chaos. The surprising properties of scaling in Wiener space make the problem of rigorously connecting this formula with the usual definition of the analytic Feynman integral a subtle one. One of the main tools in carrying this out is our definition of the 'natural extension' of $ p$th homogeneous chaos in terms of the 'scale-invariant lifting' of $ p$-forms on the white noise space $ {L^2}({\mathbb{R}_ + })$ connected with Wiener space. The key result in our development says that if $ {f_p}$ is a symmetric function in $ {L^2}(\mathbb{R}_ + ^p)$ and $ {\psi _p}({f_p})$ is the associated $ p$-form on $ {L^2}({\mathbb{R}_ + })$, then $ {\psi _p}({f_p})$ has a scaled $ {L^2}$-lifting if and only if the '$ k$th limiting trace' of $ {f_p}$ exists for $ k = 0,1, \ldots ,[p/2]$. This necessary and sufficient condition for the lifting of a $ p$-form on white noise space to a random variable on Wiener space is a worthwhile contribution to white noise theory apart from any connection with the Feynman integral since $ p$-forms play a role in white noise calculus analogous to the role played by $ p$th homogeneous chaos in Wiener calculus.

Various $ k$-traces arise naturally in this subject; we study some of their properties and relationships. The limiting $ k$-trace plays the most essential role for us.

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Keywords: Homogeneous chaos, multiple Wiener-Itô integral, scaling on Wiener space, $ p$-forms on Hilbert space, liftings, scale invariant liftings, scale invariant $ {\mathcal{L}^2}$ liftings, $ k$-trace, limiting $ k$-trace, iterated $ k$-trace, white noise, canonical Gauss measure, accessible random variables, natural extension of a multiple Wiener-Itô integral, Feynman integral, abstract Wiener space
Article copyright: © Copyright 1993 American Mathematical Society