{
  "id": "0906.0627",
  "version": "v1",
  "published": "2009-06-03T01:13:34.000Z",
  "updated": "2009-06-03T01:13:34.000Z",
  "title": "Uniqueness of values of Aronsson operators and running costs in \"tug-of-war\" games",
  "authors": [
    "Yifeng Yu"
  ],
  "comment": "To appear in \"Ann. Inst. H. Poincare Anal. Non Lineaire\"",
  "categories": [
    "math.AP"
  ],
  "abstract": "Let $A_H$ be the Aronsson operator associated with a Hamiltonian $H(x,z,p).$ Aronsson operators arise from $L^\\infty$ variational problems, two person game theory, control problems, etc. In this paper, we prove, under suitable conditions, that if $u\\in W^{1,\\infty}_{\\rm loc}(\\Omega)$ is simultaneously a viscosity solution of both of the equations $A_H(u)=f(x)$ and $A_H(u)=g(x)$ in $\\Omega$, where $f, g\\in C(\\Omega),$ then $f=g.$ The assumption $u\\in W_{loc}^{1,\\infty}(\\Omega)$ can be relaxed to $u\\in C(\\Omega)$ in many interesting situations. Also, we prove that if $f,g,u\\in C(\\Omega)$ and $u$ is simultaneously a viscosity solution of the equations ${\\Delta_\\infty u\\over |Du|^2}=-f(x)$ and ${\\Delta_{\\infty}u\\over |Du|^2}=-g(x)$ in $\\Omega$ then $f=g.$ This answers a question posed in Peres, Schramm, Scheffield and Wilson [PSSW] concerning whether or not the value function uniquely determines the running cost in the \"tug-of-war\" game.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-06-03T01:13:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J70"
    ],
    "keywords": [
      "running cost",
      "tug-of-war",
      "uniqueness",
      "viscosity solution",
      "aronsson operators arise"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009AnIHP..26.1299Y"
    }
  }
}