  Let $\psi:{\mathcal{D}}\rightarrow{\mathbf{R}}$ be a harmonic function such
that $\Delta\psi(x)=0$ for all $x\in\mathcal{D}\subset{\mathbf{R}}^{n}$. There
are then many well-established classical results:the Dirichlet problem and
Poisson formula, Harnack inequality, the Maximum Principle, the Mean Value
Property etc. Here, a 'noisy' or random domain is one for which there also
exists a classical scalar Gaussian random field (GRF) ${\mathscr{J}(x)}$
defined for all $x\in{\mathcal{D}}$ or $x\in\partial {\mathcal{D}}$ with
respect to a probability space $[\Omega,\mathcal{F},{\mathrm{I\!P}}]$. The GRF
has vanishing mean value $\mathbf{E}[\![\mathscr{J}(x)]\!] = 0$ and a regulated
covariance ${{\mathbf{E}}}[\![{{\mathscr{J}}(x)} \otimes {{\mathscr{J}}(y)}]\!]
= \alpha J(x,y;\xi)$ for all $(x,y)\in{\mathcal{D}}$ and/or
$(x,y)\in{\partial\mathcal{D}}$, with correlation length $\xi$ and
${{\mathbf{E}}}[\![{{\mathscr{J}}(x)} \otimes {{\mathscr{J}}}(x)]\!] =
\alpha<\infty$. The gradient $\nabla{{\mathscr{J}}(x)}$ and integral
$\int_{{\mathcal{D}}}{\mathscr{J}}(x) d\mu(x)$ also exist on
${\mathcal{D}}\bigcup\partial\mathcal{D}$. Harmonic functions and potentials
can become randomly perturbed GRFs of the form
$\overline{\psi(x)}=\psi(x)+\lambda{{\mathscr{J}}}(x)$. Physically, this
scenario arises from noisy sources or random fluctuations in mass/charge
density, noisy or random boundary/surface data; and introducing
turbulence/randomness into smooth fluid flows, steady state diffusions or heat
flow. This leads to stochastic modifications of classical theorems for randomly
perturbed harmonic functions and Riesz and Newtonian potentials; and to
stability estimates and bounds for the growth and decay of their volatility and
moments.
