Exact computation of Feynman-type integrals involving Gaussian random fields (Q1567432)

The authors explicitly calculate some Feynman-type integrals of the form \[ I(A)=\int_H \exp\big\{i[(h,h)-(h,Ah)]/2\big\} {\mathcal D}(h), \] where \(H\) is a real Hilbert space equipped with the canonical Gaussian measure, \(A\) is a linear self-adjoint operator on \(H\), and \({\mathcal D}(h)\) is a ``uniform measure'' on \(H\) normalized so that \(I(0)=\int_H \exp\{i(h,h)/2\} {\mathcal D}(h)\) has a prescribed value. The integral \(I(A)\) is interpreted as \(I(0)K_g(-i)\), where \(K_g(z)\) (\(z\) pure imaginary) is the Feynman integral over \(H\) of the function \(g(h)=\exp\{-i(h,Ah)/2\}\). Several cases are examined: the Brownian tent, standard Dirichlet form, and discretized two-parameter Gaussian processes (Brownian sheets).