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

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 f on Wiener space in terms of the expansion of f 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 p th homogeneous chaos in terms of the ’scale-invariant lifting’ of p p -forms on the white noise space L 2 ( R + ) {L^2}({\mathbb {R}_ + }) connected with Wiener space. The key result in our development says that if f p {f_p} is a symmetric function in L 2 ( R + p ) {L^2}(\mathbb {R}_ + ^p) and ψ p ( f p ) {\psi _p}({f_p}) is the associated p p -form on L 2 ( R + ) {L^2}({\mathbb {R}_ + }) , then ψ

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