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Let $\psi:{\mathcal{D}}\rightarrow{\mathbf{R}}$ be a harmonic function suchthat $\Delta\psi(x)=0$ for all $x\in\mathcal{D}\subset{\mathbf{R}}^{n}$. Thereare then many well-established classical results:the Dirichlet problem andPoisson formula, Harnack inequality, the Maximum Principle, the Mean ValueProperty etc. Here, a 'noisy' or random domain is one for which there alsoexists a classical scalar Gaussian random field (GRF) ${\mathscr{J}(x)}$defined for all $x\in{\mathcal{D}}$ or $x\in\partial {\mathcal{D}}$ withrespect to a probability space $[\Omega,\mathcal{F},{\mathrm{I\!P}}]$. The GRFhas vanishing mean value $\mathbf{E}[\![\mathscr{J}(x)]\!] = 0$ and a regulatedcovariance ${{\mathbf{E}}}[\![{{\mathscr{J}}(x)} \otimes {{\mathscr{J}}(y)}]\!]= \alpha J(x,y;\xi)$ 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 potentialscan become randomly perturbed GRFs of the form$\overline{\psi(x)}=\psi(x)+\lambda{{\mathscr{J}}}(x)$. Physically, thisscenario arises from noisy sources or random fluctuations in mass/chargedensity, noisy or random boundary/surface data; and introducingturbulence/randomness into smooth fluid flows, steady state diffusions or heatflow. This leads to stochastic modifications of classical theorems for randomlyperturbed harmonic functions and Riesz and Newtonian potentials; and tostability estimates and bounds for the growth and decay of their volatility andmoments.

Harmonic Functions And Linear Elliptic Dirichlet Problems With Random  Boundary Values--Stochastic Extensions Of Some Classical Theorems And  Estimates