Cylindrical Wiener process

A Cylindrical Wiener process on a real separable Hilbert space $U$ is a linear family of jointly Gaussian processes $W_t(h)$, indexed by $h\in U$, with continuous trajectories andLinearity can be understood in the space of square-integrable random variables. For a unit vector $h$, $W_t(h)$ is standard Brownian motion; orthogonal directions give independent Brownian motions. The definition specifies all finite collections of observations without requiring a $U$-valued random position.^[ref]

For an orthonormal basis $(e_k)$, let ${\beta }_k$ be independent standard Brownian motions. Thenconverges in mean square for each $h\in U$, since its variance is $t\Vert h{\Vert }^2$. The formal vector ${\sum }_k{\beta }_k(t)e_k$ is different. Its first $N$ coordinates have expected squared norm $Nt$; for fixed $t>0$, the strong law of large numbers gives $N^{-1}{\sum }_{k=1}^N{\beta }_k(t{)}^2\to t$ almost surely. The squared norm therefore diverges in infinitely many dimensions. Coordinate observations exist even though the proposed random vector does not belong to $U$.

A Hilbert-Schmidt operator $A:U\to H$ repairs this mismatch: ${\sum }_k{\beta }_k(t)A{e}_k$ converges as an $H$-valued square-integrable random variable when ${\sum }_k\Vert A{e}_k{\Vert }_H^2$ is finite.^[ref] For example, on ${\ell }^2$, setting $A{e}_k=k^{-1}{e}_k$ gives expected squared norm $t{\sum }_k{k}^{-2}=t{\pi }^2/6$. This smoothing makes an actual random position, while retaining infinitely many noisy coordinates. It is useful when defining stochastic partial differential equations and their noise terms.