Symbolic Calculus of the Wiener Process and Wiener-Hermite Functionals

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.

Stochastic Oscillators with Quadratic Nonlinearity Using WHEP and HPM Methods

“…The homotopy introduces ontinuously deform ution for the case of p = 0, , to the case of p = 1, inal ation (30) motopy ethod which is to deform continuously a simple problem (and easy to solve) into the difficult problem under study [35]. The basic assumption of the HPM method is that the solution of the original Equation (29) .…”

“…Among them, the WHEP technique was introduced in [22] using the perturbation technique to solve perturbed nonlinear problems. The WHE method utilizes the Wiener-Hermite polynomials which are the elements of a complete set of statistically orthogonal random functions [30]. The WienerHermite polynomial…”

“…There are a lot of applications in boundary value problems [24,25] and generally in different mathematical studies [26][27][28][29]. The WHE properties and description of its usage are given in [30].…”

Solution of Nonlinear Stochastic Langevin’s Equation Using WHEP, Pickard and HPM Methods

“…Due to the completeness of the Wiener-Hermite set [15], any arbitrary stochastic process ( ) ; t υ ω can be expanded as:…”

“…Since Cameron and Martin's work, WHE has become a useful tool in stochastic analysis involving white noise (Brownian motion) [14]. Also, another formulation was suggested and applied by Meecham and his co-workers [15,16]. They have developed a theory of turbulence involving a truncated WHE of the velocity field.…”