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 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.