{
  "id": "2005.02556",
  "version": "v1",
  "published": "2020-05-06T01:57:38.000Z",
  "updated": "2020-05-06T01:57:38.000Z",
  "title": "Harmonic Functions and Riesz-Newtonian Potentials on 'Noisy' or Random Domains:Stochastic Extensions of Some Classical Theorems and Estimates",
  "authors": [
    "Steven D Miller"
  ],
  "categories": [
    "math-ph",
    "math.MP",
    "math.PR"
  ],
  "abstract": "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-05-06T01:57:38.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "harmonic function",
      "random domain",
      "classical theorems",
      "stochastic extensions",
      "riesz-newtonian potentials"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}