The stochastic heat equation and convolutions

Let A generate a strongly continuous semigroup S on H and let W be cylindrical on U. For a bounded deterministic amplitude B from U to H, the zero-initial-data convolution is $Z_t=\int_0^tS(t-r)B\,dW_r$ when the time integral of $\|S(r)B\|_{\mathrm{HS}}^2$ is finite. This condition concerns each fixed time t. Continuity of the H-valued path requires an additional argument; it should not be silently inferred merely from finiteness at individual times.

For the Dirichlet Laplacian on the interval from zero to pi, choose normalized eigenfunctions with eigenvalues minus k squared. With identity noise amplitude, each coordinate is an independent Ornstein–Uhlenbeck convolution. Its variance is the integral of the squared exponential response, equal to one minus exp of minus twice k squared t, divided by twice k squared. The singularity of the full Hilbert–Schmidt norm near time zero is integrable in this one-dimensional setting.

Define the spectral Sobolev norm by $\|z\|_{H_A^s}^2=\sum_k\lambda_k^s|z_k|^2$ for positive Dirichlet eigenvalues λk. At a fixed positive time, the high-frequency heat variances behave like $1/(2\lambda_k)$. On a smooth bounded d-dimensional domain, Weyl eigenvalue growth therefore gives finite expected squared spectral Sobolev norm exactly when $s<1-d/2$ for identity cylindrical forcing. The endpoint diverges. In particular the solution is L²-valued in dimension one, but not in dimensions two or higher under this white spatial covariance.

This is a fixed-time Gaussian Hilbert-space regularity threshold, including negative spectral spaces. In the diagonal Gaussian setting, the same trace criterion also determines almost-sure membership. It is not a statement that no distribution-valued solution exists above the L² dimensional threshold. Nor does it identify pointwise Hölder exponents by ordinary Sobolev embedding alone. Finer path regularity needs increment estimates and modification theorems, and nonlinear equations can require still more careful solution spaces.

On the interval, reducing the noise amplitude in mode k by k to the power minus a changes its variance tail to order k to the power minus twice a minus two. The spectral regularity threshold improves to s less than a plus one-half. The amplitude itself is Hilbert–Schmidt only when a exceeds one-half, so heat smoothing can produce a state even outside that stronger input condition. The distinction is between the amplitude's own summability and the integrated evolved amplitude's summability.

For path continuity one useful sufficient linear criterion strengthens the operator integral with a small singular time weight: integrating r to the minus twice alpha times the squared Hilbert–Schmidt norm for some alpha strictly between zero and one-half supports a factorization argument. Gaussian moments then supply the needed time-integrability exponent. In the one-dimensional white-forced heat case, such a choice exists with alpha below one-quarter. These estimates concern the entire convolution process, unlike a single spectral trace calculation at a fixed time.