Cylindrical Wiener processes and Hilbert-space noise

Let U be a real separable Hilbert space with orthonormal basis $\{e_k\}$ and independent standard Brownian motions $\beta_k$. For each h in U, define $W_t(h)=\sum_k\langle h,e_k\rangle\beta_k(t)$ as a limit in scalar L². The sum converges because the squared coefficients are summable. It is linear as an L²-valued map in h, jointly Gaussian, and has covariance $(s\wedge t)\langle h,g\rangle_U$. It is not automatically evaluation against a U-valued random variable.

The formal vector sum of all Brownian coordinates has identity covariance. In an infinite-dimensional Hilbert space that covariance has infinite trace, so it cannot be the covariance of a Radon Gaussian random vector there. This is a structural obstruction, not a poor choice of basis. At a fixed positive time, the squared norm of the first N coordinates divided by N converges almost surely to that time, by the ordinary law of large numbers. The partial sums therefore cannot converge in U.

For a bounded nonnegative self-adjoint trace-class operator Q on U, its eigenbasis expansion gives $W^Q_t=\sum_k\sqrt{q_k}\beta_k(t)e_k$. This converges in L² with values in U, and has a continuous U-valued version. Its covariance at time t is tQ, and its increments are independent Gaussian vectors with the corresponding covariance differences. Zero eigenvalues simply suppress directions. Positive semidefinite, rather than strictly positive, covariance is sufficient.

The expected squared U-norm is time multiplied by the covariance trace. In the diagonal setting, summability of the eigenvalues is exactly the requirement for this Hilbert-valued Gaussian construction. A covariance with eigenvalues decreasing to zero can still fail the trace condition; compactness alone is weaker than trace class. Conversely, a cylindrical process can be realized in a larger space through a Hilbert–Schmidt embedding, but that changes the space in which the paths live and does not repair convergence in the original U.

A stochastic PDE can have a well-defined state even when its input noise is only cylindrical. The evolution operator may smooth each recent noise increment enough for their integrated contributions to converge. This will require an operator norm calculation, not a declaration that the noise itself has become a state-space Brownian motion. The Hilbert space used to index the noise and the Hilbert space used for the solution should therefore be named separately, even when both are represented by square-summable coordinates.

Finite-dimensional truncations are legitimate approximations only with a specified convergence topology. Every truncation has a finite covariance trace, so a successful simulation at one cutoff cannot establish an infinite-dimensional limit. One should examine how expected norm, spatial regularity and increments change with the cutoff. A cylindrical family also does not give independent point values of a spatial field: point evaluation may not even be continuous on the chosen function space. Scalar tests are the safe definition until stronger regularity has been proved.