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

Abstract

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.