= Stochastic heat equation
A <stochastic heat equation> combines diffusive smoothing with random forcing. On the real line, a basic additive model is
$$
\partial_tu=\kappa\partial_x^2u+\sigma\xi,\qquad \kappa>0,
$$
where $\xi$ 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> is
$$
u(t,x)=\sigma\int_0^t\int_{\mathbb R}G_\kappa(t-s,x-y)\,W(ds,dy),\qquad
G_\kappa(t,x)=\frac{e^{-x^2/(4\kappa t)}}{\sqrt{4\pi\kappa t}}.
$$
Here $W$ denotes the Gaussian noise integrated over space and time, and $G_\kappa$ is the <heat kernel>.<An Introduction to Stochastic PDEs by Martin Hairer>[Section 2.3.1, equations 2.5–2.6, additive white-noise model and heat-kernel representation]{ref}
The <Itô isometry> turns a rough random field into a tractable calculation:
$$
\operatorname{Var}(u(t,x))=\sigma^2\int_0^t\int_{\mathbb R}G_\kappa(s,y)^2\,dy\,ds
=\sigma^2\int_0^t\frac{ds}{\sqrt{8\pi\kappa s}}
=\sigma^2\sqrt{\frac{t}{2\pi\kappa}}.
$$
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.<An Introduction to Stochastic PDEs by Martin Hairer>[Sections 2.3.1–2.3.2, covariance calculation and dimension-dependent roughness]{ref}
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