the first chaos and the RKHS isometry
Attached to any centered Gaussian process there are TWO Hilbert spaces that turn out to be the same space in disguise: the reproducing kernel Hilbert space (a space of deterministic functions, built from the covariance) and the first chaos (a space of random variables, built from the process itself). The clean isometry between them is the engine that lets you do geometry with Gaussian random variables — angles between random variables become inner products of functions.
The first chaos (or first Wiener chaos, or the Gaussian Hilbert space H_1) is the closed linear span, inside L^2 of the underlying probability space, of all the centered Gaussian variables X_t themselves: finite linear combinations sum a_i X_(t_i), and their L^2 limits. Every element of H_1 is again a centered Gaussian variable, and the L^2 inner product is E[ZW] = Cov(Z, W). The key fact: the map sending the kernel function K_t in the RKHS to the random variable X_t in H_1 extends to a linear ISOMETRY between the RKHS H_K and the first chaos H_1, because both inner products agree on generators — < K_s, K_t >_(H_K) = K(s, t) = Cov(X_s, X_t) = < X_s, X_t >_(L^2). So a deterministic function h in the RKHS corresponds to a unique Gaussian random variable, often written < h, . > or W(h), with E[W(h) W(g)] = < h, g >_(H_K).
Why it matters: this isometry is the precise sense in which 'the covariance is a geometry.' It is the structural heart of the isonormal Gaussian process and the Wiener integral (where the first chaos is exactly the set of Wiener integrals of deterministic functions), and it underlies Cameron-Martin: the RKHS element h and its paired chaos variable W(h) are two faces of the same admissible shift. An honest caveat: the isometry lives only at the FIRST chaos — products and nonlinear functionals of Gaussians spill into higher chaoses (the Wiener chaos decomposition), where linearity and the simple isometry no longer apply.
For Brownian motion, the first chaos is the set of Wiener integrals { integral_0^1 f(t) dB_t : f in L^2[0, 1] }, all centered Gaussian with E[(integral f dB)^2] = integral f^2 dt (the Ito isometry). The RKHS element h(t) = integral_0^t f(u) du is paired with the chaos variable integral_0^1 f dB, and ||h||_(RKHS)^2 = integral f^2 dt = E[(integral f dB)^2] — the two spaces measure the same length.
RKHS function h and Gaussian variable W(h) are isometric images of each other; for Brownian motion the pairing is h(t) = integral_0^t f and integral f dB.
The isometry is linear and lives only in the first chaos; nonlinear functions of Gaussians (e.g. X_t^2) sit in higher Wiener chaoses where this clean correspondence breaks.