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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]

A mild solution of an evolution equation satisfies an integrated equation built from its linear evolution, rather than requiring all derivatives in the differential equation to exist pointwise. Let $A$ generate a strongly continuous strongly continuous semigroup $S(t)$, and consider $du=(Au+F(u))dt+Qd{W}_t$. Whenever the integrals are defined, its mild formulation isThe final term is a stochastic convolution. The linear evolution propagates both the initial condition and each later contribution for the remaining time. This is the infinite-dimensional counterpart of variation of constants.^[ref]

A deterministic heat equation already shows why the definition is useful. Take $A=\kappa {\partial }_x^2$ on $L^2(\mathbb{R})$, zero forcing, and the discontinuous initial function $u_0=1_{[0,1]}$. The mild solution is its convolution with the heat kernel:It is smooth for $t>0$ and approaches $u_0$ in $L^2$ as $t$ decreases to zero, even though the initial function lacks the derivatives appearing in the classical equation.

The word mild does not mean approximate. It specifies how the equation is interpreted. Existence, uniqueness and whether a mild solution is also a classical or distributional solution depend on the generator, initial data, nonlinear term and noise.

A stochastic convolution propagates random forcing through a linear evolution. Let $S(t)$ be a strongly continuous semigroup on a Hilbert space $H$, let $W$ be a Cylindrical Wiener process on a Hilbert space $U$, and let $Q:U\to H$ be a bounded linear operator. The expressionis an Itô integral whenever its integrand is square integrable in the Hilbert-Schmidt operator norm. In particular,This Itô isometry tests whether the accumulated forcing is actually $H$-valued. The factor $S(t-s)$ evolves the contribution arriving at time $s$ through the remaining time $t-s$. For $du=Audt+Qd{W}_t$ with deterministic initial value $u_0$, the mild solution is $u(t)=S(t)u_0+Z(t)$.^[ref]

For $S(t)=e^{-at}$, $Q=\sigma$ and real Brownian motion $B$, with $a>0$,This is the zero-initial-value Ornstein-Uhlenbeck process: damping limits the variance despite continued forcing. With $a=0$, the limit is ${\sigma }^{2}t$ instead. These formulas follow by integrating ${\sigma }^2{e}^{-2a(t-s)}$ in the Itô isometry.

The word convolution describes the dependence on the time difference $t-s$. It does not mean an ordinary integral against a differentiable noise function; Brownian motion requires a stochastic integral.

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]