= Space-time white noise
Unit-intensity Gaussian <space-time white noise> is a random distribution specified through its integrals against square-integrable deterministic functions. Write that integral as $W(f)$. The family is centered and jointly Gaussian, linear in $f$, and satisfies
$$
\mathbb E[W(f)W(g)]=\int f(t,x)g(t,x)\,dt\,dx.
$$
Formally one writes $\mathbb E[\xi(t,x)\xi(s,y)]=\delta(t-s)\delta(x-y)$, but $\xi(t,x)$ is not an ordinary pointwise random variable. The integrated definition gives the delta notation a precise meaning.<An Introduction to Stochastic PDEs by Martin Hairer>[Section 2.3.1, white-noise covariance; Example 4.63, equation 4.17 and its space-time extension]{ref}
For a space-time region $A$ of finite volume $|A|$, define $W(A)=W(\mathbf1_A)$. Then $\operatorname{Var}(W(A))=|A|$, and disjoint regions have independent noise integrals. The average over the region has a different scaling:
$$
\operatorname{Var}\left(\frac{W(A)}{|A|}\right)=\frac1{|A|}.
$$
Shrinking the averaging region increases the variance of the average. This prevents taking a finite-variance pointwise limit.
For an independent numerical example in one space dimension, a rectangular cell has volume $\Delta t\,\Delta x$. Its noise average can be sampled as $Z/\sqrt{\Delta t\,\Delta x}$ with $Z\sim N(0,1)$. In normalized coordinates, $\Delta t=0.01$ and $\Delta x=0.1$ give standard deviation about $31.62$. Giving every refined grid cell the same variance would change the continuum noise model.
Convolving <space-time white noise> with a <heat kernel> produces the random forcing contribution to a <stochastic heat equation>. Whether that smoothed field has pointwise values depends on dimension and the smoothing operator; the noise definition alone does not imply a continuous signal.
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