Wick algebra, diffusions, and computations

Abstract

We summarize pertinent mathematical results in Chap. 2,= and in this chapter, we focus on more useful derivations based on the Wick products with the aim of developing a consistent mathematical machinery for computations. Central to these developments are the Hermite polynomials, and they are used in the Wick products related to the Itô diffusions. The idea of using the Itô diffusions to model the stochastic variables that participate in the Wick type equations is inspired by the fact that many physical variables have drift and diffusive components to them. The forces and flows governed by the physical laws drive the average behaviors in noisy environments and the fluctuations in intrinsic and extrinsic forces and flows give rise to the “noise” in the system. Therefore, it is reasonable to assume that the drift term represents the average behavior, and the diffusion term is a result of the fluctuations. Another important assumption is that the effects of average behaviors and the fluctuations are additive, not multiplicative, and to simplify the analysis, we assume the coefficients associated with these terms are constants