the Wiener chaos decomposition
/ VEE-ner /
Once you have a Gaussian noise W, what do ALL square-integrable functionals of it look like? The Wiener chaos decomposition is the answer: it is the Fourier analysis of Gaussian randomness. It says the entire space L^2 of functionals measurable with respect to W splits into an orthogonal sum of pieces of increasing 'degree' — the n-th Wiener chaos — built from Hermite polynomials of the Gaussian. It is the infinite-dimensional analogue of writing a function as a power series, with Hermite polynomials playing the role of monomials.
Let W be an isonormal Gaussian process over H. The n-th Wiener chaos H_n is the closed span of all random variables H_n(W(h)) where H_n is the n-th Hermite polynomial (H_0 = 1, H_1(x) = x, H_2(x) = x^2 - 1, H_3(x) = x^3 - 3x, ...) and ||h||_H = 1. These chaoses are mutually orthogonal in L^2, and the decomposition theorem (Wiener-Ito) states that L^2(sigma(W)) = the orthogonal direct sum of H_0, H_1, H_2, .... So every square-integrable F = F(W) has a unique expansion F = sum over n of F_n with F_n in the n-th chaos, and E[F^2] = sum ||F_n||^2 (Pythagoras). The reason Hermite polynomials appear is that they are exactly the orthogonal polynomials for the Gaussian measure: E[H_m(Z) H_n(Z)] = 0 for m != n and = n! for m = n, when Z is standard normal. Each chaos H_n is moreover isometric (up to n!) to the symmetric n-fold tensor power of H, and its elements are the n-fold Wiener-Ito integrals.
Why it matters: this decomposition is the foundation of Malliavin calculus, the fourth-moment theorems for normal approximation of chaos variables, the analysis of polynomial functionals of Gaussians, and the spectral theory of the Ornstein-Uhlenbeck semigroup (each chaos H_n is an eigenspace with eigenvalue -n). A common misconception to dispel: the 'chaos' here is unrelated to dynamical-systems chaos; it is Wiener's name for the homogeneous polynomial layers. And the decomposition needs F to be in L^2 — it is an orthogonal expansion in L^2 norm, not a pointwise identity.
Let Z be standard normal (the W of a one-dimensional H). Then Z^2 = (Z^2 - 1) + 1 = H_2(Z) + H_0(Z): the deterministic part 1 sits in the 0th chaos, the centered part Z^2 - 1 in the 2nd chaos, and there is no 1st-chaos component. Squaring a Gaussian creates a degree-2 term — exactly why E[Z^2] = 1 (the H_0 piece) and Var(Z^2) = E[H_2^2] = 2.
Hermite polynomials are the orthogonal building blocks: Z^2 splits as H_2(Z) + H_0(Z) into the 2nd and 0th Wiener chaoses.
Wiener 'chaos' is just the name for the Hermite/polynomial layers; it has nothing to do with deterministic chaos, and the expansion is an L^2 (not pointwise) identity.