= Stochastic partial differential equation
A <stochastic partial differential equation> describes an unknown random field through spatial differential operators and random forcing. A model form is $du=(\kappa\Delta u+F(u))\,dt+G(u)\,dW_t$. The meaning of the noise, boundary data and solution is part of the specification; writing a formal derivative of <Brownian motion> does not define an ordinary function-valued forcing term.<An Introduction to Stochastic PDEs by Martin Hairer>[Introduction and mild solutions of linear stochastic evolution equations]{ref}
A simple exact example avoids the complications of noise at every spatial point. On $0<x<1$ with zero boundary values, consider
$$
du=\kappa u_{xx}\,dt+\sin(\pi x)\,dB_t,\qquad u(0,x)=0.
$$
The solution is $u(t,x)=Y_t\sin(\pi x)$, where
$$
Y_t=\int_0^te^{-\kappa\pi^2(t-s)}\,dB_s,\qquad\operatorname{Var}(Y_t)=\frac{1-e^{-2\kappa\pi^2t}}{2\kappa\pi^2}.
$$
Here $\kappa>0$. Spatial diffusion damps a single mode while noise continually excites it. More spatially irregular forcing can yield solutions that are distributions rather than pointwise functions, so regularity must be proved for the chosen dimension and noise.
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