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

Let $\psi :\mathcal{D}\to \mathbf{R}$ be a harmonic function such that $\mathrm{\Delta }\psi (x)=0$ 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) $\mathcal{J}(x)$ defined for all $x\in \mathcal{D}$ or $x\in \mathrm{\partial }\mathcal{D}$ with respect to a probability space $[\mathrm{\Omega },\mathcal{F},\mathrm{I}\text{\hspace{0.17em}⁣}\mathrm{P}]$. The GRF has vanishing mean value $\mathbf{E}[\text{\hspace{0.17em}⁣}[\mathcal{J}(x)]\text{\hspace{0.17em}⁣}]=0$ and a regulated covariance ${{$\mathbf{E}$}}[![{{\mathscr{J}}(x)} \otimes {{\mathscr{J}}(y)}]!] = \alpha J(x,y;\xi)$forall$(x,y)\in{$\mathcal{D}$}$and\mathrm{/}or$(x,y)\in{\partial$\mathcal{D}$}$,withcorrelationlength$\xi$and${{$\mathbf{E}$}}[![{{\mathscr{J}}(x)} \otimes {{\mathscr{J}}}(x)]!] = \alpha<\infty$\mathrm{.}Thegradient$\nabla{{\mathscr{J}}(x)}$andintegral$\int_{{$\mathcal{D}$}}{\mathscr{J}}(x) d\mu(x)$alsoexiston${$\mathcal{D}$}\bigcup\partial$\mathcal{D}$$. Harmonic functions and potentials can become randomly perturbed GRFs of the form $\overline{\psi (x)}=\psi (x)+\lambda \mathcal{J}(x)$. 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.