Space-time white noise and scaling

On space-time $\mathbb R\times\mathbb R^d$, real white noise ξ is a centered Gaussian family indexed linearly by $L^2$ test functions, with covariance equal to their L² inner product. For a measurable set E with $0<|E|<\infty$, ξ evaluated on its indicator has variance equal to its volume. The average over E divides that random variable by the volume, giving variance equal to the inverse volume. Disjoint-set integrals are independent because they are jointly Gaussian with zero covariance.

Overlapping-set averages are correlated in proportion to the intersection volume, with their normalizing volumes included. There is no finite-variance point value obtained by shrinking a set indefinitely. White in time also differs from a random potential that is fixed in time and rough only in space. Such noises have different scaling dimensions and lead to different singularity thresholds, even if both are denoted by the same letter in informal equations. State which covariance is being used before counting powers.

Under the parabolic dilation $(t,x)\mapsto(\lambda^2t,\lambda x)$, volume scales by $\lambda^{d+2}$. The formally rescaled field $\lambda^{(d+2)/2}\xi(\lambda^2t,\lambda x)$ has the same white-noise law, with that notation interpreted through tests rather than point evaluation. An L¹-normalized parabolic mollifier at scale ε therefore has squared L² norm proportional to ε to the minus d minus two. Its white-noise evaluation has standard deviation proportional to ε to the minus half of d plus two.

In local parabolic Hölder–Besov terminology, white noise has regularity strictly below minus half of d plus two. A heat convolution gains two parabolic orders, suggesting response regularity strictly below one minus d/2. This description uses a different norm from the spectral Hilbert Sobolev norm earlier, even when the numerical threshold coincides. For these linear Gaussian models, identifying Hölder regularity rigorously requires the relevant test-function or increment estimates, not merely an algebraic dimensional assignment.

For a semilinear heat equation with polynomial drift of degree p greater than one and additive space-time white noise, assign the linear field nominal regularity alpha equal to one minus d/2. Formal multiplication gives degree-p homogeneity p alpha, and heat integration adds two. A favorable local power count requires p alpha plus two to exceed alpha, or d less than two plus four divided by p minus one. For a cubic drift this predicts subcritical spatial dimensions below four.

Negative-regularity fields do not acquire a classical product merely because the resulting power count is favorable. One must construct enhanced stochastic objects, select counterterms and prove stability in an appropriate solution space. At the critical equality there is no positive regularity improvement in this elementary count; the subcritical theory cannot simply be applied unchanged. Derivative nonlinearities, conservation laws, multiplicative noise and non-parabolic equations require their own counts. A scaling calculation is useful precisely when its limited role remains explicit.