Harmonic Functions and Riesz-Newtonian Potentials on 'Noisy' or Random Domains:Stochastic Extensions of Some Classical Theorems and Estimates

Steven D Miller

Published 2020-05-06Version 1

Let $\psi:{\mathcal{D}}\rightarrow{\mathrm{R}}$ be a harmonic function such that $\Delta \psi(x)=0$ for all $x\in\mathcal{D}\subset{\mathrm{R}}^{n}$. 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) ${\mathrm{I\!F}(x)}$ defined for all $x\in{\mathcal{D}}$ or $x\in\partial {\mathcal{D}}$ with respect to a probability space $[\Omega,\mathscr{F},{\mathsf{P}}]$. The GRF has vanishing mean value $\mathsf{E}[\![\mathrm{I\!F}(x)]\!] = 0$ and a regulated covariance ${{\mathsf{E}}}[\![{{\mathrm{I\!F}}(x)} \otimes {{\mathrm{I\!F}}(y)}]\!] = \alpha K(x,y;\xi)$ for all $(x,y)\in{\mathcal{D}}$, with correlation length $\xi$ and ${{\mathsf{E}}}[\![{{\mathrm{I\!F}}(x)} \otimes {{\mathrm{I\!F}}}(x)]\!] = \alpha<\infty$. The gradient $\nabla{{\mathrm{I\!F}}(x)}$ and integrals $\int_{{\mathcal{D}}}{\mathrm{I\!F}}(x) d\mu(x)$ also exist on ${\mathcal{D}}$. Harmonic functions and potentials can become randomly perturbed GRFs of the form $\overline{\psi(x)}=\psi(x)+\lambda{{\mathrm{I\!F}}}(x)$. 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 for the growth and decay of their volatility and moments.