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Martin Hairer is a mathematician whose work joins probability theory and analysis. He developed regularity structures to give precise meaning to classes of singular stochastic partial differential equations. His EPFL biography identifies this work as central to his 2014 Fields Medal.^[ref]

The motivating obstacle is that a noisy solution can be too irregular for the nonlinear operations appearing in its own equation. Smoothing the input makes an ordinary equation possible, but the resulting solutions need not converge when the smoothing is removed. Regularity structures replace ordinary polynomial local expansions with expansions tailored to the singular input and organize the necessary corrections. The theory proves convergence of appropriate corrected approximations for the equations it covers.^[ref]

His An Introduction to Stochastic PDEs by Martin Hairer gives a complementary starting point through the stochastic heat equation and mild solutions. Integrating the heat evolution against a random driving term clarifies why solving a noisy equation requires specifying both the evolution operator and the interpretation of the noise.^[ref]

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]