Noncommutative operations on Wiener functionals and Feynman's operational calculus (Q1114140)

The article [Mem. Am. Math. Soc. 351, 78 p. (1986; Zbl 0638.28009)] gave a rigorous formulation of the heuristic operational calculus due to \textit{R. P. Feynman} [Phys. Rev. 84, 108-128 (1951; Zbl 0044.233)]. The present paper pursues further Feynman's ideas by introducing new noncommutative operations \({\dot+}\) and \(*\) on Wiener functionals, and show, in particular, two important equalities, \[ K_{\lambda}^{s+t}(F*G)=K^ s_{\lambda}(F)K^ t_{\lambda}(G)\quad and\quad \exp (F\dot +G)=\exp (F)*\exp (G), \] where the function \(K^ t_{\lambda}\) is defined by \[ (K^ t_{\lambda}(F)\psi)\xi)=\int_{C^ t_ 0}F(\lambda^{-1/2} x+\xi)\psi (\lambda^{-1/2}x(t)+\xi)dm_ t(x) \] and \(m_ t\) means Wiener measure on Wiener space \(C^ t_ 0\). A shortened form of this paper appears in C. R. Acad. Sci., Paris, Sér. I 304, 523-526 (1987; Zbl 0609.46024).