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Generalized Fourier-Feynman transforms, convolution products, and first variations on function space. (English) Zbl 1202.60133
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 generalized analytic Fourier-Feynman transform, the generalized convolution product and the 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.

MSC:
60J65 Brownian motion
28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)
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