Graded rings of homogeneous chaos generated by polynomials in noises depending on time and space parameters, respectively

We take two kinds of noises; one is Gaussian noise and the other is that depending on the intensities which are determined by the law of small probabilities. As for the Gaussian case, the polynomials are continuous versions of those in finitely many $\dot B(t)$'s, where $B(t)$ is a Brownian motion. We may compare those polynomials with generalized white noise functionals (Hida distributions) of finite degree. As for the other case, polynomials are formed by idealized elemental random variables parametrized by the intensity, so that we need different technique from Gaussian case in order to discuss the analysis on polynomials in question.