Symbolic Calculus of the Wiener Process and Wiener-Hermite Functionals

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Abstract
A new definition is given for the ``ideal random function'' (derivative of the Wiener function), which separates out infinite factors by fullest exploitation of the possibilities of the Dirac delta function. By allowing all integrals to be written formally as sums, this facilitates the definition and manipulation of the Wiener‐Hermite functionals, especially for vector random processes of multiple argument. Expansion of a random function in Wiener‐Hermite functionals is discussed. An expression is derived for the expectation value of the product of any number of Wiener‐Hermite functionals; this is all that is needed in principle to obtain full statistical information from the Wiener‐Hermite functional expansion of a random function. The method is illustrated by the calculation of the first correction to the flatness factor (measure of Gaussianity) of a nearly‐Gaussian random function.

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