We show a method and the structure to calculate the characteristic functions of quadratic Wiener functionals by using classical Weierstrass-Hadamard's theory on entire functions. We also examine the idea by an example for Gaussian processes with multiple Markovian property.
"Finite dimensional determinants as characteristic functions of quadratic Wiener functionals." Electron. Commun. Probab. 9 26 - 35, 2004. https://doi.org/10.1214/ECP.v9-1091