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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 satisfiesFormally 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.^[ref]

For a space-time region $A$ of finite volume $|A|$, define $W(A)=W(1_A)$. Then $\mathrm{Var}(W(A))=|A|$, and disjoint regions have independent noise integrals. The average over the region has a different scaling: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.

A stochastic heat equation combines diffusive smoothing with random forcing. On the real line, a basic additive model iswhere $\xi$ is space-time white noise, formally specified by $\mathbb{E}[\xi (t,x)\xi (s,y)]=\delta (t-s)\delta (x-y)$. The noise is a random distribution, so this expression is interpreted through integration, not ordinary pointwise values. With zero initial data the mild solution isHere $W$ denotes the Gaussian noise integrated over space and time, and $G_{\kappa }$ is the heat kernel.^[ref]

The Itô isometry turns a rough random field into a tractable calculation:Thus the standard deviation at a fixed location grows like $t^{1/4}$ from zero. Diffusion reduces the influence of each earlier disturbance while fresh noise continually arrives.

Dimension changes the answer. In $d$ spatial dimensions the kernel's squared integral is proportional to $s^{-d/2}$. Its integral near zero is finite only for $d<2$. Consequently this additive white-noise model has pointwise finite-variance values in one dimension, whereas higher-dimensional versions require a distribution-valued interpretation or spatially smoother noise. The formal differential expression alone does not settle the solution's regularity.^[ref]