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 $f$ 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.