the isonormal Gaussian process and white noise
The isonormal Gaussian process is the cleanest, most abstract Gaussian object: instead of indexing the process by time t, you index it by the elements h of a Hilbert space H, in such a way that the geometry of H is copied exactly into the covariance. White noise is the concrete special case where H = L^2 of some measure space, and it is the random 'derivative of Brownian motion' that does not exist as an ordinary function but does exist as this Hilbert-indexed Gaussian family.
Precisely, given a separable Hilbert space H, an isonormal Gaussian process is a centered Gaussian family W = (W(h))_(h in H) with E[W(g) W(h)] = < g, h >_H. So W is a linear isometry from H into the first chaos: it turns inner products of vectors into covariances of Gaussian variables, and orthogonal vectors into independent Gaussians. When H = L^2(R_+) and we write W(1_[0, t]) =: B_t, the resulting B is exactly Brownian motion, because E[B_s B_t] = < 1_[0, s], 1_[0, t] > = min(s, t). The map h -> W(h) is then the Wiener integral, W(f) = integral f dB, and the noise W itself is white noise: a Gaussian random measure assigning to disjoint sets independent centered Gaussians whose variances are the sizes of the sets. White noise is not a process with values at single points; only its integrals against L^2 test functions are honest random variables.
Why it matters: the isonormal picture strips away the index set and keeps only the linear-algebraic essence, so theorems proved once (Wiener chaos decomposition, Malliavin calculus, the Ornstein-Uhlenbeck semigroup) apply to ALL Gaussian processes simultaneously via their RKHS. The caveat is exactly the singular nature of white noise: dB/dt does not exist pointwise, integral f dB is defined for f in L^2 but the 'sample path' of white noise is a distribution, not a function — which is precisely why stochastic integration (Ito) rather than ordinary integration is required.
Take H = L^2[0, 1] and an orthonormal basis e_1, e_2, .... Then xi_n := W(e_n) are independent standard normals, and W(h) = sum < h, e_n > xi_n. Reconstructing Brownian motion: B_t = W(1_[0, t]) = sum (integral_0^t e_n) xi_n — the random Fourier-type series whose increments are independent because orthogonal indicators map to independent Gaussians.
An isonormal process copies H's geometry into covariances; choosing H = L^2 makes W white noise and W(1_[0,t]) Brownian motion.
White noise has no pointwise values: only integrals integral f dB (f in L^2) are random variables; the 'derivative of Brownian motion' exists as a distribution, not a function.