Generalized Fourier-Feynman transforms, convolution products, and first variations on function space
Let \(Y\) be a generalized Brownian motion process, i.e.\ a Gaussian process with an absolutely continuous mean function \(a(t)\), \(t\in [0,T]\), with \(a(0)=0\), \(a'\in L^2[0,T]\), and covariance function \(r(s,t)=\min\{b(s),b(t)\}\), \(s,t\in [0,T]\) where \(b\) is a strictly increasing \(C^1\)-function with \(b(0)=0\). The authors are interested in functionals of the form \[ F(x) = f(\langle\alpha_1,x \rangle, \ldots, \langle\alpha_n,x \rangle) \] where \(\langle\alpha,x \rangle = \int_0^T \alpha(t)\,dx(t)\) is the Paley-Wiener-Zygmund stochastic integral for \(x\in C_{a,b}[0,T]\). (Unfortunately, the authors fail to define this function space in the paper). The authors introduce the \textit{generalized analytic Fourier-Feynman transform}, the \textit{generalized convolution product} and the \textit{generalized first variation} for such functionals \(F\). The main part of the paper is devoted to the study of various relationships if any two or any three distinct operations are combined.
scientific article; zbMATH DE number 5760093